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$$\frac{2 x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{1}{\cos^{2}{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -69.1078034322536$$
$$x_{2} = 15.6760783451944$$
$$x_{3} = 50.2555336325565$$
$$x_{4} = 18.8229989180076$$
$$x_{5} = 12.5264763376692$$
$$x_{6} = -75.3915917440781$$
$$x_{7} = -100.525991117835$$
$$x_{8} = -47.1132774827275$$
$$x_{9} = 59.6818828624266$$
$$x_{10} = 94.2424741940464$$
$$x_{11} = 31.4000043168626$$
$$x_{12} = 2.97508632168828$$
$$x_{13} = 62.8238944845809$$
$$x_{14} = 72.2497107001058$$
$$x_{15} = -28.2566407733299$$
$$x_{16} = -59.6818828624266$$
$$x_{17} = -81.6752872670354$$
$$x_{18} = 84.817106677999$$
$$x_{19} = 43.9709264903445$$
$$x_{20} = 21.9683925318703$$
$$x_{21} = -6.20274981679304$$
$$x_{22} = 100.525991117835$$
$$x_{23} = 81.6752872670354$$
$$x_{24} = 56.5398246709304$$
$$x_{25} = 6.20274981679304$$
$$x_{26} = -87.9589098892909$$
$$x_{27} = 47.1132774827275$$
$$x_{28} = 34.5430455066495$$
$$x_{29} = -15.6760783451944$$
$$x_{30} = -43.9709264903445$$
$$x_{31} = -84.817106677999$$
$$x_{32} = 69.1078034322536$$
$$x_{33} = 9.37147510585595$$
$$x_{34} = 37.6858450405302$$
$$x_{35} = 91.1006985770946$$
$$x_{36} = -40.8284587489214$$
$$x_{37} = 78.5334497119428$$
$$x_{38} = 87.9589098892909$$
$$x_{39} = -21.9683925318703$$
$$x_{40} = -12.5264763376692$$
$$x_{41} = -62.8238944845809$$
$$x_{42} = -50.2555336325565$$
$$x_{43} = 28.2566407733299$$
$$x_{44} = -94.2424741940464$$
$$x_{45} = -78.5334497119428$$
$$x_{46} = -53.397711687542$$
$$x_{47} = 97.3842380053013$$
$$x_{48} = 25.1128337203766$$
$$x_{49} = -2.97508632168828$$
$$x_{50} = 65.9658661929102$$
$$x_{51} = -37.6858450405302$$
$$x_{52} = -18.8229989180076$$
$$x_{53} = 75.3915917440781$$
$$x_{54} = -97.3842380053013$$
$$x_{55} = -65.9658661929102$$
$$x_{56} = -72.2497107001058$$
$$x_{57} = -9.37147510585595$$
$$x_{58} = -34.5430455066495$$
$$x_{59} = -25.1128337203766$$
$$x_{60} = 40.8284587489214$$
$$x_{61} = -56.5398246709304$$
$$x_{62} = -31.4000043168626$$
$$x_{63} = 53.397711687542$$
$$x_{64} = -91.1006985770946$$
Signos de extremos en los puntos:
(-69.10780343225363, -69.1114209687341)
(15.676078345194368, 15.692026211395)
(50.255533632556485, 50.2605082091241)
(18.822998918007553, 18.8362805423167)
(12.5264763376692, 12.5464340650668)
(-75.39159174407808, -75.3949077637325)
(-100.52599111783519, -100.528478036862)
(-47.11327748272753, -47.1185838424919)
(59.681882862426576, 59.6860717383134)
(94.24247419404638, 94.2451269257593)
(31.400004316862624, 31.4079660992075)
(2.9750863216882792, 3.05911749691083)
(62.82389448458093, 62.8278738622055)
(72.24971070010584, 72.2531709215735)
(-28.256640773329945, -28.2654882510611)
(-59.681882862426576, -59.6860717383134)
(-81.67528726703536, -81.6783481684214)
(84.817106677999, 84.8200541966045)
(43.97092649034452, 43.9766120653359)
(21.968392531870297, 21.9797725178951)
(-6.202749816793043, -6.2430545215424)
(100.52599111783519, 100.528478036862)
(81.67528726703536, 81.6783481684214)
(56.53982467093041, 56.5442463330324)
(6.202749816793043, 6.2430545215424)
(-87.95890988929088, -87.9617521255159)
(47.11327748272753, 47.1185838424919)
(34.54304550664949, 34.5502828534536)
(-15.676078345194368, -15.692026211395)
(-43.97092649034452, -43.9766120653359)
(-84.817106677999, -84.8200541966045)
(69.10780343225363, 69.1114209687341)
(9.371475105855954, 9.3981518026594)
(37.68584504053022, 37.6924788310086)
(91.10069857709462, 91.1034427931534)
(-40.8284587489214, -40.8345819288714)
(78.53344971194282, 78.5366330688553)
(87.95890988929088, 87.9617521255159)
(-21.968392531870297, -21.9797725178951)
(-12.5264763376692, -12.5464340650668)
(-62.82389448458093, -62.8278738622055)
(-50.255533632556485, -50.2605082091241)
(28.256640773329945, 28.2654882510611)
(-94.24247419404638, -94.2451269257593)
(-78.53344971194282, -78.5366330688553)
(-53.39771168754203, -53.40239353611)
(97.38423800530128, 97.3868051558497)
(25.112833720376596, 25.1227887896764)
(-2.9750863216882792, -3.05911749691083)
(65.96586619291024, 65.9696560317228)
(-37.68584504053022, -37.6924788310086)
(-18.822998918007553, -18.8362805423167)
(75.39159174407808, 75.3949077637325)
(-97.38423800530128, -97.3868051558497)
(-65.96586619291024, -65.9696560317228)
(-72.24971070010584, -72.2531709215735)
(-9.371475105855954, -9.3981518026594)
(-34.54304550664949, -34.5502828534536)
(-25.112833720376596, -25.1227887896764)
(40.8284587489214, 40.8345819288714)
(-56.53982467093041, -56.5442463330324)
(-31.400004316862624, -31.4079660992075)
(53.39771168754203, 53.40239353611)
(-91.10069857709462, -91.1034427931534)
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:
$$x_{1} = 15.6760783451944$$
$$x_{2} = 50.2555336325565$$
$$x_{3} = 18.8229989180076$$
$$x_{4} = 12.5264763376692$$
$$x_{5} = 59.6818828624266$$
$$x_{6} = 94.2424741940464$$
$$x_{7} = 31.4000043168626$$
$$x_{8} = 2.97508632168828$$
$$x_{9} = 62.8238944845809$$
$$x_{10} = 72.2497107001058$$
$$x_{11} = 84.817106677999$$
$$x_{12} = 43.9709264903445$$
$$x_{13} = 21.9683925318703$$
$$x_{14} = 100.525991117835$$
$$x_{15} = 81.6752872670354$$
$$x_{16} = 56.5398246709304$$
$$x_{17} = 6.20274981679304$$
$$x_{18} = 47.1132774827275$$
$$x_{19} = 34.5430455066495$$
$$x_{20} = 69.1078034322536$$
$$x_{21} = 9.37147510585595$$
$$x_{22} = 37.6858450405302$$
$$x_{23} = 91.1006985770946$$
$$x_{24} = 78.5334497119428$$
$$x_{25} = 87.9589098892909$$
$$x_{26} = 28.2566407733299$$
$$x_{27} = 97.3842380053013$$
$$x_{28} = 25.1128337203766$$
$$x_{29} = 65.9658661929102$$
$$x_{30} = 75.3915917440781$$
$$x_{31} = 40.8284587489214$$
$$x_{32} = 53.397711687542$$
Puntos máximos de la función:
$$x_{32} = -69.1078034322536$$
$$x_{32} = -75.3915917440781$$
$$x_{32} = -100.525991117835$$
$$x_{32} = -47.1132774827275$$
$$x_{32} = -28.2566407733299$$
$$x_{32} = -59.6818828624266$$
$$x_{32} = -81.6752872670354$$
$$x_{32} = -6.20274981679304$$
$$x_{32} = -87.9589098892909$$
$$x_{32} = -15.6760783451944$$
$$x_{32} = -43.9709264903445$$
$$x_{32} = -84.817106677999$$
$$x_{32} = -40.8284587489214$$
$$x_{32} = -21.9683925318703$$
$$x_{32} = -12.5264763376692$$
$$x_{32} = -62.8238944845809$$
$$x_{32} = -50.2555336325565$$
$$x_{32} = -94.2424741940464$$
$$x_{32} = -78.5334497119428$$
$$x_{32} = -53.397711687542$$
$$x_{32} = -2.97508632168828$$
$$x_{32} = -37.6858450405302$$
$$x_{32} = -18.8229989180076$$
$$x_{32} = -97.3842380053013$$
$$x_{32} = -65.9658661929102$$
$$x_{32} = -72.2497107001058$$
$$x_{32} = -9.37147510585595$$
$$x_{32} = -34.5430455066495$$
$$x_{32} = -25.1128337203766$$
$$x_{32} = -56.5398246709304$$
$$x_{32} = -31.4000043168626$$
$$x_{32} = -91.1006985770946$$
Decrece en los intervalos
$$\left[100.525991117835, \infty\right)$$
Crece en los intervalos
$$\left[-2.97508632168828, 2.97508632168828\right]$$