Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada$$- 2 x e^{3} \sin{\left(2 x \right)} + e^{3} \cos{\left(2 x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 100.533451628845$$
$$x_{2} = 9.45120497843001$$
$$x_{3} = -37.7057417444241$$
$$x_{4} = 48.6998194395369$$
$$x_{5} = 65.9772348386275$$
$$x_{6} = -103.674968932212$$
$$x_{7} = 28.2831721399108$$
$$x_{8} = 23.5725488683805$$
$$x_{9} = -39.2762729921215$$
$$x_{10} = -22.0025089604154$$
$$x_{11} = 61.2651372785773$$
$$x_{12} = -80.113733189628$$
$$x_{13} = 36.1352335301545$$
$$x_{14} = -86.3966915707365$$
$$x_{15} = -81.6844695124177$$
$$x_{16} = 67.5479430595368$$
$$x_{17} = 80.113733189628$$
$$x_{18} = 37.7057417444241$$
$$x_{19} = -43.9879802762466$$
$$x_{20} = 26.7128952386973$$
$$x_{21} = 58.123765151966$$
$$x_{22} = 1.71280922974086$$
$$x_{23} = 51.8411010631448$$
$$x_{24} = -72.2600907017656$$
$$x_{25} = -36.1352335301545$$
$$x_{26} = -17.2932121076445$$
$$x_{27} = 56.5530882745116$$
$$x_{28} = 72.2600907017656$$
$$x_{29} = -15.7238573187731$$
$$x_{30} = 86.3966915707365$$
$$x_{31} = -59.6944483144154$$
$$x_{32} = 81.6844695124177$$
$$x_{33} = 70.6893712463639$$
$$x_{34} = -89.5381827032021$$
$$x_{35} = -65.9772348386275$$
$$x_{36} = 29.8535036526677$$
$$x_{37} = -50.2704553934212$$
$$x_{38} = 22.0025089604154$$
$$x_{39} = 20.432585165244$$
$$x_{40} = 50.2704553934212$$
$$x_{41} = 12.5862231633233$$
$$x_{42} = -75.4015392197413$$
$$x_{43} = -0.43016679450969$$
$$x_{44} = 3.21864908958597$$
$$x_{45} = 6.32264361192832$$
$$x_{46} = 45.5585806972324$$
$$x_{47} = -11.0182483639693$$
$$x_{48} = -73.8308134276772$$
$$x_{49} = 89.5381827032021$$
$$x_{50} = 34.5647514869476$$
$$x_{51} = -23.5725488683805$$
$$x_{52} = -51.8411010631448$$
$$x_{53} = -83.2552080991765$$
$$x_{54} = -14.154821427226$$
$$x_{55} = -97.3919391862849$$
$$x_{56} = -28.2831721399108$$
$$x_{57} = -45.5585806972324$$
$$x_{58} = 0.43016679450969$$
$$x_{59} = 114.67031200546$$
$$x_{60} = 7.88564243740794$$
$$x_{61} = -42.4173943590211$$
$$x_{62} = -3.21864908958597$$
$$x_{63} = -7.88564243740794$$
$$x_{64} = -94.2504320905443$$
$$x_{65} = 87.9674362306479$$
$$x_{66} = -64.4065309145547$$
$$x_{67} = -87.9674362306479$$
$$x_{68} = 78.542999266617$$
$$x_{69} = -9.45120497843001$$
$$x_{70} = -31.4238815972272$$
$$x_{71} = -20.432585165244$$
$$x_{72} = 15.7238573187731$$
$$x_{73} = 64.4065309145547$$
$$x_{74} = -6.32264361192832$$
$$x_{75} = -29.8535036526677$$
$$x_{76} = 94.2504320905443$$
$$x_{77} = 92.6796807176258$$
$$x_{78} = 95.8211849371972$$
$$x_{79} = 73.8308134276772$$
$$x_{80} = -67.5479430595368$$
$$x_{81} = -58.123765151966$$
$$x_{82} = 42.4173943590211$$
$$x_{83} = 59.6944483144154$$
$$x_{84} = -1.71280922974086$$
$$x_{85} = 43.9879802762466$$
$$x_{86} = 14.154821427226$$
$$x_{87} = -53.4117555918474$$
$$x_{88} = -95.8211849371972$$
$$x_{89} = -61.2651372785773$$
Signos de extremos en los puntos:
3
(100.53345162884467, 100.532208284673*e )
3
(9.451204978430011, 9.43800684898451*e )
3
(-37.705741744424074, -37.702427036601*e )
3
(48.69981943953688, -48.6972528978117*e )
3
(65.97723483862752, 65.9753403273413*e )
3
(-103.67496893221228, -103.673763262022*e )
3
(28.28317213991076, 28.2787535864381*e )
3
(23.572548868380515, -23.567247878771*e )
3
(-39.27627299212146, 39.2730907958671*e )
3
(-22.002508960415422, -21.9968299895532*e )
3
(61.2651372785773, -61.263097068409*e )
3
(-80.11373318962796, 80.112172953406*e )
3
(36.13523353015448, -36.1317747991247*e )
3
(-86.39669157073652, 86.3952447924177*e )
3
(-81.68446951241769, -81.6829392767655*e )
3
(67.54794305953683, -67.546092598104*e )
3
(80.11373318962796, -80.112172953406*e )
3
(37.705741744424074, 37.702427036601*e )
3
(-43.98798027624661, -43.9851388662124*e )
3
(26.71289523869733, -26.7082170799481*e )
3
(58.12376515196605, -58.1216146879934*e )
3
(1.7128092297408641, -1.64418569779545*e )
3
(51.84110106314479, -51.8386900171372*e )
3
(-72.26009070176562, -72.2583609016736*e )
3
(-36.13523353015448, 36.1317747991247*e )
3
(-17.29321210764446, 17.2859883667942*e )
3
(56.55308827451163, 56.5508780915478*e )
3
(72.26009070176562, 72.2583609016736*e )
3
(-15.723857318773117, -15.7159136392673*e )
3
(86.39669157073652, -86.3952447924177*e )
3
(-59.69444831441541, -59.6923544275184*e )
3
(81.68446951241769, 81.6829392767655*e )
3
(70.68937124636392, -70.687603012927*e )
3
(-89.53818270320214, 89.5367866833941*e )
3
(-65.97723483862752, -65.9753403273413*e )
3
(29.85350365266773, -29.8493174201329*e )
3
(-50.27045539342116, -50.267969027913*e )
3
(22.002508960415422, 21.9968299895532*e )
3
(20.432585165244035, -20.4264702322587*e )
3
(50.27045539342116, 50.267969027913*e )
3
(12.586223163323332, 12.5763034089358*e )
3
(-75.40153921974125, -75.3998814833205*e )
3
(-0.43016679450968986, -0.280548169095523*e )
3
(3.2186490895859734, 3.18050197241693*e )
3
(6.322643611928322, 6.30296564894634*e )
3
(45.55858069723237, -45.5558372248235*e )
3
(-11.018248363969283, 11.0069210395792*e )
3
(-73.83081342767719, 73.829120425871*e )
3
(89.53818270320214, -89.5367866833941*e )
3
(34.56475148694763, 34.5611356534609*e )
3
(-23.572548868380515, 23.567247878771*e )
3
(-51.84110106314479, 51.8386900171372*e )
3
(-83.2552080991765, 83.2537067322156*e )
3
(-14.154821427226006, 14.1459987695472*e )
3
(-97.39193918628494, -97.390655737879*e )
3
(-28.28317213991076, -28.2787535864381*e )
3
(-45.55858069723237, 45.5558372248235*e )
3
(0.43016679450968986, 0.280548169095523*e )
3
(114.67031200545999, -114.669221939379*e )
3
(7.885642437407941, -7.86983848106687*e )
3
(-42.417394359021145, 42.4144477618284*e )
3
(-3.2186490895859734, -3.18050197241693*e )
3
(-7.885642437407941, 7.86983848106687*e )
3
(-94.25043209054431, -94.2491058646707*e )
3
(87.96743623064788, 87.9660152847086*e )
3
(-64.40653091455466, 64.4045902053056*e )
3
(-87.96743623064788, -87.9660152847086*e )
3
(78.54299926661696, 78.5414078300528*e )
3
(-9.451204978430011, -9.43800684898451*e )
3
(-31.423881597227226, -31.4199044860773*e )
3
(-20.432585165244035, 20.4264702322587*e )
3
(15.723857318773117, 15.7159136392673*e )
3
(64.40653091455466, -64.4045902053056*e )
3
(-6.322643611928322, -6.30296564894634*e )
3
(-29.85350365266773, 29.8493174201329*e )
3
(94.25043209054431, 94.2491058646707*e )
3
(92.67968071762581, -92.6783320156182*e )
3
(95.82118493719717, -95.8198804506423*e )
3
(73.83081342767719, -73.829120425871*e )
3
(-67.54794305953683, 67.546092598104*e )
3
(-58.12376515196605, 58.1216146879934*e )
3
(42.417394359021145, -42.4144477618284*e )
3
(59.69444831441541, 59.6923544275184*e )
3
(-1.7128092297408641, 1.64418569779545*e )
3
(43.98798027624661, 43.9851388662124*e )
3
(14.154821427226006, -14.1459987695472*e )
3
(-53.41175559184737, -53.4094154368825*e )
3
(-95.82118493719717, 95.8198804506423*e )
3
(-61.2651372785773, 61.263097068409*e )
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:
$$x_{1} = -37.7057417444241$$
$$x_{2} = 48.6998194395369$$
$$x_{3} = -103.674968932212$$
$$x_{4} = 23.5725488683805$$
$$x_{5} = -22.0025089604154$$
$$x_{6} = 61.2651372785773$$
$$x_{7} = 36.1352335301545$$
$$x_{8} = -81.6844695124177$$
$$x_{9} = 67.5479430595368$$
$$x_{10} = 80.113733189628$$
$$x_{11} = -43.9879802762466$$
$$x_{12} = 26.7128952386973$$
$$x_{13} = 58.123765151966$$
$$x_{14} = 1.71280922974086$$
$$x_{15} = 51.8411010631448$$
$$x_{16} = -72.2600907017656$$
$$x_{17} = -15.7238573187731$$
$$x_{18} = 86.3966915707365$$
$$x_{19} = -59.6944483144154$$
$$x_{20} = 70.6893712463639$$
$$x_{21} = -65.9772348386275$$
$$x_{22} = 29.8535036526677$$
$$x_{23} = -50.2704553934212$$
$$x_{24} = 20.432585165244$$
$$x_{25} = -75.4015392197413$$
$$x_{26} = -0.43016679450969$$
$$x_{27} = 45.5585806972324$$
$$x_{28} = 89.5381827032021$$
$$x_{29} = -97.3919391862849$$
$$x_{30} = -28.2831721399108$$
$$x_{31} = 114.67031200546$$
$$x_{32} = 7.88564243740794$$
$$x_{33} = -3.21864908958597$$
$$x_{34} = -94.2504320905443$$
$$x_{35} = -87.9674362306479$$
$$x_{36} = -9.45120497843001$$
$$x_{37} = -31.4238815972272$$
$$x_{38} = 64.4065309145547$$
$$x_{39} = -6.32264361192832$$
$$x_{40} = 92.6796807176258$$
$$x_{41} = 95.8211849371972$$
$$x_{42} = 73.8308134276772$$
$$x_{43} = 42.4173943590211$$
$$x_{44} = 14.154821427226$$
$$x_{45} = -53.4117555918474$$
Puntos máximos de la función:
$$x_{45} = 100.533451628845$$
$$x_{45} = 9.45120497843001$$
$$x_{45} = 65.9772348386275$$
$$x_{45} = 28.2831721399108$$
$$x_{45} = -39.2762729921215$$
$$x_{45} = -80.113733189628$$
$$x_{45} = -86.3966915707365$$
$$x_{45} = 37.7057417444241$$
$$x_{45} = -36.1352335301545$$
$$x_{45} = -17.2932121076445$$
$$x_{45} = 56.5530882745116$$
$$x_{45} = 72.2600907017656$$
$$x_{45} = 81.6844695124177$$
$$x_{45} = -89.5381827032021$$
$$x_{45} = 22.0025089604154$$
$$x_{45} = 50.2704553934212$$
$$x_{45} = 12.5862231633233$$
$$x_{45} = 3.21864908958597$$
$$x_{45} = 6.32264361192832$$
$$x_{45} = -11.0182483639693$$
$$x_{45} = -73.8308134276772$$
$$x_{45} = 34.5647514869476$$
$$x_{45} = -23.5725488683805$$
$$x_{45} = -51.8411010631448$$
$$x_{45} = -83.2552080991765$$
$$x_{45} = -14.154821427226$$
$$x_{45} = -45.5585806972324$$
$$x_{45} = 0.43016679450969$$
$$x_{45} = -42.4173943590211$$
$$x_{45} = -7.88564243740794$$
$$x_{45} = 87.9674362306479$$
$$x_{45} = -64.4065309145547$$
$$x_{45} = 78.542999266617$$
$$x_{45} = -20.432585165244$$
$$x_{45} = 15.7238573187731$$
$$x_{45} = -29.8535036526677$$
$$x_{45} = 94.2504320905443$$
$$x_{45} = -67.5479430595368$$
$$x_{45} = -58.123765151966$$
$$x_{45} = 59.6944483144154$$
$$x_{45} = -1.71280922974086$$
$$x_{45} = 43.9879802762466$$
$$x_{45} = -95.8211849371972$$
$$x_{45} = -61.2651372785773$$
Decrece en los intervalos
$$\left[114.67031200546, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -103.674968932212\right]$$