179:@0.937671:0.974518:0.968375:0.974518:0.968375:0.956333:0.937671:0.956333:0.000000:0.000000:0.000000
Pensamiento espacial, numérico y variacional:@0.522009:0.043661:0.926913:0.043661:0.926913:0.024916:0.522009:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Ejemplo 3. :@0.073089:0.292295:0.156106:0.292295:0.156106:0.275625:0.073089:0.275625:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Apliquemos el criterio de la primera derivada para hallar los valores máximos :@0.155661:0.292295:0.710726:0.292295:0.710726:0.275861:0.155661:0.275861:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y mínimos de la función  ( ) = :@0.073086:0.308937:0.291654:0.308937:0.291654:0.292503:0.073086:0.292503:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.251158:0.308937:0.268037:0.308937:0.268037:0.292531:0.251158:0.292531:0.000000:0.000000:0.000000
x:@0.291543:0.308937:0.298998:0.308937:0.298998:0.292531:0.291543:0.292531:0.000000
3 :@0.298946:0.302873:0.306490:0.302873:0.306490:0.293292:0.298946:0.293292:0.000000:0.000000
–:@0.306438:0.309519:0.315642:0.309519:0.315642:0.291865:0.306438:0.291865:0.000000
 3 .:@0.315550:0.308937:0.338982:0.308937:0.338982:0.292503:0.315550:0.292503:0.000000:0.000000:0.000000:0.000000
x:@0.328398:0.308937:0.335853:0.308937:0.335853:0.292531:0.328398:0.292531:0.000000
Entonces:  :@0.073071:0.329157:0.151276:0.329157:0.151276:0.312723:0.073071:0.312723:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Por lo tanto, los puntos críticos son  1 y 1.:@0.073089:0.349368:0.375288:0.349368:0.375288:0.332934:0.073089:0.332934:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
–:@0.329215:0.349950:0.338418:0.349950:0.338418:0.332296:0.329215:0.332296:0.000000
Con la ayuda de una tabla podemos validar el signo de  ´( ) en los intervalos::@0.073096:0.376730:0.629185:0.376730:0.629185:0.360296:0.073096:0.360296:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f  x:@0.474135:0.376730:0.499703:0.376730:0.499703:0.360323:0.474135:0.360323:0.000000:0.000000:0.000000:0.000000
 (-∞, 1), es ( 1,1) y (1,∞).:@0.073096:0.393371:0.246382:0.393371:0.246382:0.376938:0.073096:0.376938:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
–:@0.103413:0.393954:0.112616:0.393954:0.112616:0.376300:0.103413:0.376300:0.000000
–:@0.157677:0.393954:0.166881:0.393954:0.166881:0.376300:0.157677:0.376300:0.000000
Intervalo:@0.129561:0.421008:0.200876:0.421008:0.200876:0.404061:0.129561:0.404061:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Valor de prueba:@0.246907:0.421008:0.372479:0.421008:0.372479:0.404061:0.246907:0.404061:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Signo de :@0.422954:0.421008:0.495900:0.421008:0.495900:0.404061:0.422954:0.404061:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Conclusión:@0.557621:0.421008:0.645165:0.421008:0.645165:0.404061:0.557621:0.404061:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(-∞,-1):@0.142446:0.441491:0.188041:0.441491:0.188041:0.425058:0.142446:0.425058:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x=-2:@0.293146:0.441491:0.326261:0.441491:0.326261:0.425058:0.293146:0.425058:0.000000:0.000000:0.000000:0.000000
f´(-2)=9>0:@0.420653:0.441491:0.494466:0.441491:0.494466:0.425058:0.420653:0.425058:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
crece:@0.581809:0.441491:0.621009:0.441491:0.621009:0.425058:0.581809:0.425058:0.000000:0.000000:0.000000:0.000000:0.000000
(-1,1):@0.147232:0.461975:0.183273:0.461975:0.183273:0.445541:0.147232:0.445541:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x=0:@0.295852:0.461975:0.323555:0.461975:0.323555:0.445541:0.295852:0.445541:0.000000:0.000000:0.000000
f´(0)=-3<0:@0.420653:0.461975:0.494466:0.461975:0.494466:0.445541:0.420653:0.445541:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
decrece:@0.572403:0.461975:0.630398:0.461975:0.630398:0.445541:0.572403:0.445541:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(1,∞):@0.147839:0.482458:0.182610:0.482458:0.182610:0.466024:0.147839:0.466024:0.000000:0.000000:0.000000:0.000000:0.000000
x=2:@0.295834:0.482458:0.323536:0.482458:0.323536:0.466024:0.295834:0.466024:0.000000:0.000000:0.000000
f´(2)=9>0:@0.423341:0.482458:0.491742:0.482458:0.491742:0.466024:0.423341:0.466024:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
crece:@0.581790:0.482458:0.620990:0.482458:0.620990:0.466024:0.581790:0.466024:0.000000:0.000000:0.000000:0.000000:0.000000
De la tabla anterior concluimos que hay un máximo relativo en  = 1 y un mínimo :@0.073087:0.510139:0.710420:0.510139:0.710420:0.493705:0.073087:0.493705:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.560254:0.510139:0.567709:0.510139:0.567709:0.493733:0.560254:0.493733:0.000000
2:@0.578974:0.509584:0.594307:0.509584:0.594307:0.495869:0.578974:0.495869:0.000000
relativo en  =1, como se puede ver en la gráfica de la función mostrada en la Figura 3.3. :@0.073087:0.526781:0.709108:0.526781:0.709108:0.510347:0.073087:0.510347:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.152901:0.526781:0.160356:0.526781:0.160356:0.510375:0.152901:0.510375:0.000000
Figura 3.3:@0.361141:0.553523:0.418335:0.553523:0.418335:0.540077:0.361141:0.540077:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y:@0.358874:0.581816:0.365368:0.581816:0.365368:0.568469:0.358874:0.568469:0.000000
x:@0.511757:0.747236:0.518251:0.747236:0.518251:0.733889:0.511757:0.733889:0.000000
2:@0.457332:0.734531:0.463513:0.734531:0.463513:0.723620:0.457332:0.723620:0.000000
4:@0.523538:0.734531:0.529719:0.734531:0.529719:0.723620:0.523538:0.723620:0.000000
-2:@0.311716:0.734531:0.321595:0.734531:0.321595:0.723620:0.311716:0.723620:0.000000:0.000000
0:@0.379162:0.734531:0.385343:0.734531:0.385343:0.723620:0.379162:0.723620:0.000000
10:@0.372054:0.667967:0.384416:0.667967:0.384416:0.657056:0.372054:0.657056:0.000000:0.000000
20:@0.372054:0.606441:0.384416:0.606441:0.384416:0.595530:0.372054:0.595530:0.000000:0.000000
(1,-2):@0.410349:0.754798:0.443972:0.754798:0.443972:0.740250:0.410349:0.740250:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(-1,2):@0.301244:0.706385:0.334835:0.706385:0.334835:0.691837:0.301244:0.691837:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Máximo relativo:@0.299534:0.684756:0.383294:0.684756:0.383294:0.673663:0.299534:0.673663:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Mínimo relativo:@0.400608:0.768976:0.482260:0.768976:0.482260:0.757883:0.400608:0.757883:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y = x:@0.483442:0.578086:0.510567:0.578086:0.510567:0.564739:0.483442:0.564739:0.000000:0.000000:0.000000:0.000000:0.000000
3 :@0.510566:0.574556:0.519301:0.574556:0.519301:0.563645:0.510566:0.563645:0.000000:0.000000
+3:@0.519301:0.578086:0.535632:0.578086:0.535632:0.564750:0.519301:0.564750:0.000000:0.000000
Criterio de la primera derivada:@0.086924:0.145472:0.316992:0.145472:0.316992:0.128803:0.086924:0.128803:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Consideremos a  como un punto crítico de una función continua  en un intervalo :@0.086924:0.165688:0.690393:0.165688:0.690393:0.149255:0.086924:0.149255:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
c :@0.207712:0.165688:0.218609:0.165688:0.218609:0.149282:0.207712:0.149282:0.000000:0.000000
f :@0.568458:0.165688:0.576391:0.165688:0.576391:0.149282:0.568458:0.149282:0.000000:0.000000
abierto ( ,  ) con     ( ,  ). Si es derivable en ( ,  ), excepto quizás en  ,  ( ) puede :@0.086942:0.182330:0.692099:0.182330:0.692099:0.165896:0.086942:0.165896:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a b:@0.147705:0.182330:0.173217:0.182330:0.173217:0.165924:0.147705:0.165924:0.000000:0.000000:0.000000
c:@0.213474:0.182330:0.221131:0.182330:0.221131:0.165924:0.213474:0.165924:0.000000
∈:@0.225015:0.182913:0.240348:0.182913:0.240348:0.165258:0.225015:0.165258:0.000000
a b:@0.249000:0.182330:0.274512:0.182330:0.274512:0.165924:0.249000:0.165924:0.000000:0.000000:0.000000
 f :@0.298534:0.182330:0.309707:0.182330:0.309707:0.165924:0.298534:0.165924:0.000000:0.000000:0.000000
a b:@0.427863:0.182330:0.453376:0.182330:0.453376:0.165924:0.427863:0.165924:0.000000:0.000000:0.000000
c f c:@0.600137:0.182330:0.631871:0.182330:0.631871:0.165924:0.600137:0.165924:0.000000:0.000000:0.000000:0.000000:0.000000
clasificarse como sigue::@0.086869:0.198972:0.256381:0.198972:0.256381:0.182538:0.086869:0.182538:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
• :@0.113725:0.219188:0.122229:0.219188:0.122229:0.202754:0.113725:0.202754:0.000000:0.000000
Si  ´ cambia de positiva a negativa en  ,  ( ) es un mínimo relativo de  .:@0.143925:0.219188:0.651450:0.219188:0.651450:0.202754:0.143925:0.202754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f :@0.160105:0.219188:0.168039:0.219188:0.168039:0.202782:0.160105:0.202782:0.000000:0.000000
c f c:@0.419261:0.219188:0.450995:0.219188:0.450995:0.202782:0.419261:0.202782:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.643627:0.219188:0.648321:0.219188:0.648321:0.202782:0.643627:0.202782:0.000000
• :@0.113725:0.235830:0.122229:0.235830:0.122229:0.219396:0.113725:0.219396:0.000000:0.000000
Si  ´ cambia de negativa a positiva en  ,  ( ) es un máximo relativo de  .:@0.143925:0.235830:0.653990:0.235830:0.653990:0.219396:0.143925:0.219396:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f :@0.160105:0.235830:0.168039:0.235830:0.168039:0.219424:0.160105:0.219424:0.000000:0.000000
c f c:@0.419261:0.235830:0.450995:0.235830:0.450995:0.219424:0.419261:0.219424:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.646167:0.235830:0.650861:0.235830:0.650861:0.219424:0.646167:0.219424:0.000000
• :@0.113725:0.252472:0.122229:0.252472:0.122229:0.236038:0.113725:0.236038:0.000000:0.000000
Si  ´ no cambia de signo en c, entonces   no es un valor extremo de  .:@0.143925:0.252472:0.643830:0.252472:0.643830:0.236038:0.143925:0.236038:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f :@0.160105:0.252472:0.168039:0.252472:0.168039:0.236066:0.160105:0.236066:0.000000:0.000000
f:@0.433729:0.252472:0.438423:0.252472:0.438423:0.236066:0.433729:0.236066:0.000000
f:@0.636007:0.252472:0.640700:0.252472:0.640700:0.236066:0.636007:0.236066:0.000000
Criterio de la segunda derivada:@0.088888:0.819723:0.325134:0.819723:0.325134:0.803053:0.088888:0.803053:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Supongamos que   es un punto crítico de una :@0.088888:0.839942:0.475587:0.839942:0.475587:0.823509:0.088888:0.823509:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
c:@0.236869:0.839942:0.244527:0.839942:0.244527:0.823536:0.236869:0.823536:0.000000
función   en un intervalo abierto ( ,  ) con :@0.088888:0.856584:0.395277:0.856584:0.395277:0.840150:0.088888:0.840150:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.148178:0.856584:0.152871:0.856584:0.152871:0.840178:0.148178:0.840178:0.000000
a b:@0.329949:0.856584:0.355186:0.856584:0.355186:0.840178:0.329949:0.840178:0.000000:0.000000:0.000000
c :@0.394890:0.856584:0.405787:0.856584:0.405787:0.840178:0.394890:0.840178:0.000000:0.000000
∈ :@0.405511:0.857167:0.425354:0.857167:0.425354:0.839512:0.405511:0.839512:0.000000:0.000000
( ,  ) y :@0.424949:0.856584:0.475532:0.856584:0.475532:0.840150:0.424949:0.840150:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a b:@0.429753:0.856584:0.454990:0.856584:0.454990:0.840178:0.429753:0.840178:0.000000:0.000000:0.000000
supongamos que  ´( ) = 0.:@0.088851:0.873226:0.282366:0.873226:0.282366:0.856792:0.088851:0.856792:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f  c:@0.220941:0.873226:0.246712:0.873226:0.246712:0.856820:0.220941:0.856820:0.000000:0.000000:0.000000:0.000000
• :@0.088900:0.893438:0.097404:0.893438:0.097404:0.877004:0.088900:0.877004:0.000000:0.000000
Si  \( ) < 0:@0.119102:0.893438:0.467005:0.893438:0.467005:0.877004:0.119102:0.877004:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f  x:@0.135282:0.893438:0.160868:0.893438:0.160868:0.877032:0.135282:0.877032:0.000000:0.000000:0.000000:0.000000
f  c:@0.200352:0.893438:0.220692:0.893438:0.220692:0.877032:0.200352:0.877032:0.000000:0.000000:0.000000:0.000000
f:@0.455200:0.893438:0.459894:0.893438:0.459894:0.877032:0.455200:0.877032:0.000000
Si  \( )>0:@0.119021:0.910080:0.452506:0.910080:0.452506:0.893646:0.119021:0.893646:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f  x:@0.135201:0.910080:0.160787:0.910080:0.160787:0.893674:0.135201:0.893674:0.000000:0.000000:0.000000:0.000000
f  c:@0.192374:0.910080:0.212714:0.910080:0.212714:0.893674:0.192374:0.893674:0.000000:0.000000:0.000000:0.000000
f:@0.444682:0.910080:0.449376:0.910080:0.449376:0.893674:0.444682:0.893674:0.000000
Criterio de concavidad:@0.526061:0.819396:0.696853:0.819396:0.696853:0.802726:0.526061:0.802726:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Sea   una función derivable dos veces en un intervalo :@0.526061:0.839616:0.919247:0.839616:0.919247:0.823182:0.526061:0.823182:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.556046:0.839616:0.560740:0.839616:0.560740:0.823210:0.556046:0.823210:0.000000
abierto ( ,  ).:@0.526042:0.856258:0.620251:0.856258:0.620251:0.839824:0.526042:0.839824:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a b:@0.586805:0.856258:0.612317:0.856258:0.612317:0.839851:0.586805:0.839851:0.000000:0.000000:0.000000
• :@0.526089:0.880042:0.534593:0.880042:0.534593:0.863608:0.526089:0.863608:0.000000:0.000000
Si  \( ) > 0:@0.556291:0.880042:0.713024:0.880042:0.713024:0.863608:0.556291:0.863608:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f  x:@0.572563:0.880042:0.598242:0.880042:0.598242:0.863636:0.572563:0.863636:0.000000:0.000000:0.000000:0.000000
x    a b f :@0.712974:0.880042:0.792512:0.880042:0.792512:0.863636:0.712974:0.863636:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∈:@0.723687:0.880624:0.739020:0.880624:0.739020:0.862970:0.723687:0.862970:0.000000
( ,  ),  es cóncava hacia :@0.742260:0.880042:0.919146:0.880042:0.919146:0.863608:0.742260:0.863608:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
arriba en ( ,  ).  :@0.556199:0.896683:0.670278:0.896684:0.670278:0.880250:0.556199:0.880250:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a b:@0.628890:0.896683:0.654402:0.896683:0.654402:0.880277:0.628890:0.880277:0.000000:0.000000:0.000000
• :@0.526089:0.913326:0.534593:0.913326:0.534593:0.896892:0.526089:0.896892:0.000000:0.000000
Si  \( ) < 0:@0.556291:0.913326:0.713024:0.913326:0.713024:0.896892:0.556291:0.896892:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f  x:@0.572563:0.913326:0.598242:0.913326:0.598242:0.896919:0.572563:0.896919:0.000000:0.000000:0.000000:0.000000
x    a b f :@0.712974:0.913326:0.792512:0.913326:0.792512:0.896919:0.712974:0.896919:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∈:@0.723687:0.913908:0.739020:0.913908:0.739020:0.896254:0.723687:0.896254:0.000000
( ,  ),  es cóncava hacia :@0.742260:0.913326:0.919146:0.913326:0.919146:0.896892:0.742260:0.896892:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
abajo en ( ,  ). :@0.556199:0.929967:0.665207:0.929967:0.665207:0.913534:0.556199:0.913534:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a b:@0.627730:0.929967:0.653242:0.929967:0.653242:0.913561:0.627730:0.913561:0.000000:0.000000:0.000000
Negocios y economía:@0.740294:0.214768:0.888603:0.214768:0.888603:0.199614:0.740294:0.199614:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
: :@0.888460:0.214768:0.895037:0.214768:0.895037:0.199828:0.888460:0.199828:0.000000:0.000000
Al aplicar el criterio de :@0.740299:0.229897:0.889744:0.229897:0.889744:0.214957:0.740299:0.214957:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
la primera derivada, se :@0.740299:0.245026:0.891660:0.245026:0.891660:0.230086:0.740299:0.230086:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
puede identificar el punto :@0.740299:0.260155:0.915696:0.260155:0.915696:0.245215:0.740299:0.245215:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
donde una función de :@0.740299:0.275284:0.891623:0.275284:0.891623:0.260344:0.740299:0.260344:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
costos o ingresos alcanza :@0.740299:0.290413:0.910263:0.290413:0.910263:0.275473:0.740299:0.275473:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
su máximo o mínimo, lo :@0.740299:0.305542:0.903163:0.305542:0.903163:0.290602:0.740299:0.290602:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
que ayuda a maximizar :@0.740299:0.320670:0.896185:0.320670:0.896185:0.305731:0.740299:0.305731:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ganancias o minimizar :@0.740299:0.335799:0.892040:0.335799:0.892040:0.320860:0.740299:0.320860:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
costos.:@0.740299:0.350928:0.785433:0.350928:0.785433:0.335989:0.740299:0.335989:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
En ingeniería:@0.740299:0.366057:0.830608:0.366057:0.830608:0.350903:0.740299:0.350903:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
: la primera :@0.830494:0.366057:0.906945:0.366057:0.906945:0.351117:0.830494:0.351117:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
derivada ayuda a modelar :@0.740299:0.381186:0.914987:0.381186:0.914987:0.366246:0.740299:0.366246:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
la velocidad de cambio de :@0.740299:0.396315:0.917773:0.396315:0.917773:0.381375:0.740299:0.381375:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
variables como la tensión :@0.740299:0.411444:0.911672:0.411444:0.911672:0.396504:0.740299:0.396504:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
o la deformación de un :@0.740299:0.426573:0.897768:0.426573:0.897768:0.411633:0.740299:0.411633:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
material, para predecir :@0.740299:0.441702:0.891295:0.441702:0.891295:0.426762:0.740299:0.426762:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
cómo responderá bajo :@0.740299:0.456831:0.894277:0.456831:0.894277:0.441891:0.740299:0.441891:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
diferentes fuerzas. :@0.740299:0.471960:0.862307:0.471960:0.862307:0.457020:0.740299:0.457020:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
En biomecánica:@0.740299:0.638378:0.849788:0.638378:0.849788:0.623224:0.740299:0.623224:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
, en :@0.849671:0.638378:0.876797:0.638378:0.876797:0.623438:0.849671:0.623438:0.000000:0.000000:0.000000:0.000000:0.000000
la modelación de :@0.740299:0.653507:0.858457:0.653507:0.858457:0.638567:0.740299:0.638567:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
crecimiento poblacional :@0.740299:0.668636:0.904194:0.668636:0.904194:0.653696:0.740299:0.653696:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
o de otros fenómenos :@0.740299:0.683765:0.889821:0.683765:0.889821:0.668825:0.740299:0.668825:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
naturales, el criterio de la :@0.740299:0.698894:0.907259:0.698894:0.907259:0.683954:0.740299:0.683954:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
primera derivada identifica :@0.740299:0.714023:0.920610:0.714023:0.920610:0.699083:0.740299:0.699083:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
cuándo un cambio es :@0.740299:0.729152:0.887289:0.729152:0.887289:0.714212:0.740299:0.714212:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
positivo o negativo.:@0.740299:0.744281:0.869549:0.744281:0.869549:0.729341:0.740299:0.729341:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Interdisciplinariedad :@0.740294:0.150157:0.906668:0.150157:0.906668:0.133210:0.740294:0.133210:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000