Para hallar los extremos hay que resolver la ecuación
dxdf(x)=0(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
dxdf(x)=primera derivadacos(x1)cos(x)+x2sin(x1)sin(x)=0Resolvermos esta ecuaciónRaíces de esta ecuación
x1=73.8274298446286x2=−1.77032184622602x3=−45.5531040577872x4=−48.6946947926006x5=−89.5353920205745x6=−64.4026531424855x7=−7.85605530705307x8=−73.8274298446286x9=−32.986750731248x10=−51.8362859646913x11=−61.2610610949586x12=76.9690222061413x13=−67.5442452975689x14=−26.7035900960034x15=1.77032184622602x16=89.5353920205745x17=58.1194691856341x18=−54.9778774562622x19=10.9963284351197x20=−95.818577071243x21=39.2699246862074x22=−76.9690222061413x23=95.818577071243x24=−98.9601696199687x25=70.6858375373694x26=83.2522070532694x27=−39.2699246862074x28=23.5620213951938x29=202.632726276733x30=−36.1283367275698x31=64.4026531424855x32=1983.91576074208x33=92.6769845372215x34=51.8362859646913x35=61.2610610949586x36=−70.6858375373694x37=−92.6769845372215x38=−86.393799524576x39=48.6946947926006x40=−42.4115139342614x41=−58.1194691856341x42=45.5531040577872x43=67.5442452975689x44=−10.9963284351197x45=17.278953653359x46=4.72203085083256x47=80.1106146116865x48=−20.4204697787209x49=−83.2522070532694x50=−80.1106146116865x51=−17.278953653359x52=20.4204697787209x53=54.9778774562622x54=32.986750731248x55=7.85605530705307x56=29.8451678396463x57=−4.72203085083256x58=86.393799524576x59=36.1283367275698x60=98.9601696199687x61=−23.5620213951938x62=26.7035900960034x63=14.1375214322216x64=−29.8451678396463x65=42.4115139342614x66=−14.1375214322216Signos de extremos en los puntos:
(73.82742984462863, -0.999908266517767)
(-1.7703218462260202, -0.827901301432779)
(-45.55310405778716, -0.999759055668925)
(-48.69469479260059, 0.99978914130114)
(-89.53539202057446, -0.999937629960711)
(-64.40265314248548, -0.999879453727449)
(-7.856055307053065, -0.991907384132263)
(-73.82742984462863, 0.999908266517767)
(-32.98675073124796, -0.999540529076331)
(-51.83628596469129, -0.999813924651387)
(-61.26106109495865, 0.99986677327201)
(76.9690222061413, 0.999915602036213)
(-67.5442452975689, 0.999890406351476)
(-26.703590096003428, -0.999298898656899)
(1.7703218462260202, 0.827901301432779)
(89.53539202057446, 0.999937629960711)
(58.11946918563406, 0.999851981482606)
(-54.97787745626216, 0.999834582238175)
(10.996328435119693, -0.995867574424807)
(-95.81857707124303, -0.999945541379844)
(39.269924686207396, 0.999675789868926)
(-76.9690222061413, -0.999915602036213)
(95.81857707124303, 0.999945541379844)
(-98.9601696199687, 0.999948944157013)
(70.68583753736935, 0.999899931368176)
(83.25220705326942, 0.999927860474679)
(-39.269924686207396, -0.999675789868926)
(23.56202139519381, -0.99909950536129)
(202.63272627673345, 0.99998782272965)
(-36.128336727569845, 0.999616957824081)
(64.40265314248548, 0.999879453727449)
(1983.9157607420825, -0.999999872964957)
(92.6769845372215, -0.999941786728398)
(51.83628596469129, 0.999813924651387)
(61.26106109495865, -0.99986677327201)
(-70.68583753736935, -0.999899931368176)
(-92.6769845372215, 0.999941786728398)
(-86.39379952457602, 0.999933011536799)
(48.69469479260059, -0.99978914130114)
(-42.41151393426139, 0.999722039894878)
(-58.11946918563406, -0.999851981482606)
(45.55310405778716, 0.999759055668925)
(67.5442452975689, -0.999890406351476)
(-10.996328435119693, 0.995867574424807)
(17.278953653359007, -0.998325755891758)
(4.722030850832559, -0.977614273674147)
(80.11061461168646, -0.999922091606656)
(-20.42046977872091, -0.998801179244391)
(-83.25220705326942, -0.999927860474679)
(-80.11061461168646, 0.999922091606656)
(-17.278953653359007, 0.998325755891758)
(20.42046977872091, 0.998801179244391)
(54.97787745626216, -0.999834582238175)
(32.98675073124796, 0.999540529076331)
(7.856055307053065, 0.991907384132263)
(29.845167839646347, -0.999438717024592)
(-4.722030850832559, 0.977614273674147)
(86.39379952457602, -0.999933011536799)
(36.128336727569845, -0.999616957824081)
(98.9601696199687, -0.999948944157013)
(-23.56202139519381, 0.99909950536129)
(26.703590096003428, 0.999298898656899)
(14.13752143222161, 0.997499348016665)
(-29.845167839646347, 0.999438717024592)
(42.41151393426139, -0.999722039894878)
(-14.13752143222161, -0.997499348016665)
Intervalos de crecimiento y decrecimiento de la función:Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=73.8274298446286x2=−1.77032184622602x3=−45.5531040577872x4=−89.5353920205745x5=−64.4026531424855x6=−7.85605530705307x7=−32.986750731248x8=−51.8362859646913x9=−26.7035900960034x10=10.9963284351197x11=−95.818577071243x12=−76.9690222061413x13=−39.2699246862074x14=23.5620213951938x15=1983.91576074208x16=92.6769845372215x17=61.2610610949586x18=−70.6858375373694x19=48.6946947926006x20=−58.1194691856341x21=67.5442452975689x22=17.278953653359x23=4.72203085083256x24=80.1106146116865x25=−20.4204697787209x26=−83.2522070532694x27=54.9778774562622x28=29.8451678396463x29=86.393799524576x30=36.1283367275698x31=98.9601696199687x32=42.4115139342614x33=−14.1375214322216Puntos máximos de la función:
x33=−48.6946947926006x33=−73.8274298446286x33=−61.2610610949586x33=76.9690222061413x33=−67.5442452975689x33=1.77032184622602x33=89.5353920205745x33=58.1194691856341x33=−54.9778774562622x33=39.2699246862074x33=95.818577071243x33=−98.9601696199687x33=70.6858375373694x33=83.2522070532694x33=202.632726276733x33=−36.1283367275698x33=64.4026531424855x33=51.8362859646913x33=−92.6769845372215x33=−86.393799524576x33=−42.4115139342614x33=45.5531040577872x33=−10.9963284351197x33=−80.1106146116865x33=−17.278953653359x33=20.4204697787209x33=32.986750731248x33=7.85605530705307x33=−4.72203085083256x33=−23.5620213951938x33=26.7035900960034x33=14.1375214322216x33=−29.8451678396463Decrece en los intervalos
[1983.91576074208,∞)Crece en los intervalos
(−∞,−95.818577071243]