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$$\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)}} - \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}\right) \tan^{\cos{\left(x \right)}}{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -99.2788581438821$$
$$x_{2} = -45.8717830328556$$
$$x_{3} = -86.712487529523$$
$$x_{4} = -77.2877095687536$$
$$x_{5} = -17.5974491505475$$
$$x_{6} = -96.1372654902923$$
$$x_{7} = 10.6768847317606$$
$$x_{8} = 1.25210677099125$$
$$x_{9} = -74.1461169151638$$
$$x_{10} = 23.2432553461198$$
$$x_{11} = -11.3142638433679$$
$$x_{12} = 32.6680333068892$$
$$x_{13} = 76.6503304571463$$
$$x_{14} = -61.5797463008046$$
$$x_{15} = -20.7390418041373$$
$$x_{16} = 26.3848479997096$$
$$x_{17} = -36.4470050720863$$
$$x_{18} = 45.2344039212484$$
$$x_{19} = 4.39369942458105$$
$$x_{20} = -23.8806344577271$$
$$x_{21} = 13.8184773853504$$
$$x_{22} = -1.88948588259854$$
$$x_{23} = -30.1638197649067$$
$$x_{24} = 16.9600700389402$$
$$x_{25} = 54.6591818820177$$
$$x_{26} = -55.296560993625$$
$$x_{27} = 95.499886378685$$
$$x_{28} = 42.0928112676586$$
$$x_{29} = -52.1549683400352$$
$$x_{30} = -89.8540801831128$$
$$x_{31} = -80.4293022223434$$
$$x_{32} = 51.5175892284279$$
$$x_{33} = 64.0839598427871$$
$$x_{34} = 89.2167010715055$$
$$x_{35} = 82.9335157643259$$
$$x_{36} = 57.8007745356075$$
$$x_{37} = 35.809625960479$$
$$x_{38} = -67.8629316079842$$
$$x_{39} = 60.9423671891973$$
$$x_{40} = -42.7301903792658$$
$$x_{41} = 48.3759965748382$$
$$x_{42} = 86.0751084179157$$
$$x_{43} = -49.0133756864454$$
$$x_{44} = -33.3054124184965$$
$$x_{45} = -39.5885977256761$$
$$x_{46} = 38.9512186140688$$
$$x_{47} = -5.03107853618833$$
$$x_{48} = 20.10166269253$$
$$x_{49} = -71.004524261574$$
$$x_{50} = 98.6414790322748$$
$$x_{51} = -64.7213389543944$$
$$x_{52} = 7.53529207817084$$
$$x_{53} = 70.3671451499667$$
$$x_{54} = 79.7919231107361$$
$$x_{55} = 67.2255524963769$$
$$x_{56} = 92.3582937250953$$
$$x_{57} = -27.0222271113169$$
$$x_{58} = -83.5708948759332$$
$$x_{59} = -8.17267118977813$$
Signos de extremos en los puntos:
(-99.27885814388213, 1.41542505382344)
(-45.871783032855646, 0.706501553931617)
(-86.71248752952296, 1.41542505382344)
(-77.28770956875358, 0.706501553931617)
(-17.597449150547508, 1.41542505382344)
(-96.13726549029234, 0.706501553931617)
(10.676884731760632, 0.706501553931617)
(1.252106770991253, 1.41542505382344)
(-74.14611691516379, 1.41542505382344)
(23.243255346119806, 0.706501553931617)
(-11.31426384336792, 1.41542505382344)
(32.668033306889185, 1.41542505382344)
(76.6503304571463, 1.41542505382344)
(-61.579746300804615, 1.41542505382344)
(-20.7390418041373, 0.706501553931617)
(26.3848479997096, 1.41542505382344)
(-36.44700507208626, 1.41542505382344)
(45.23440392124836, 1.41542505382344)
(4.393699424581047, 0.706501553931617)
(-23.880634457727094, 1.41542505382344)
(13.818477385350427, 1.41542505382344)
(-1.8894858825985401, 0.706501553931617)
(-30.16381976490668, 1.41542505382344)
(16.96007003894022, 0.706501553931617)
(54.65918188201774, 0.706501553931617)
(-55.29656099362502, 1.41542505382344)
(95.49988637868505, 1.41542505382344)
(42.09281126765857, 0.706501553931617)
(-52.15496834003523, 0.706501553931617)
(-89.85408018311276, 0.706501553931617)
(-80.42930222234337, 1.41542505382344)
(51.517589228427944, 1.41542505382344)
(64.08395984278712, 1.41542505382344)
(89.21670107150547, 1.41542505382344)
(82.93351576432588, 1.41542505382344)
(57.80077453560753, 1.41542505382344)
(35.80962596047898, 0.706501553931617)
(-67.8629316079842, 1.41542505382344)
(60.94236718919733, 0.706501553931617)
(-42.73019037926585, 1.41542505382344)
(48.375996574838155, 0.706501553931617)
(86.07510841791567, 0.706501553931617)
(-49.013375686445436, 1.41542505382344)
(-33.30541241849647, 0.706501553931617)
(-39.58859772567606, 0.706501553931617)
(38.95121861406877, 1.41542505382344)
(-5.031078536188334, 1.41542505382344)
(20.101662692530013, 1.41542505382344)
(-71.00452426157399, 0.706501553931617)
(98.64147903227484, 0.706501553931617)
(-64.72133895439441, 0.706501553931617)
(7.53529207817084, 1.41542505382344)
(70.3671451499667, 1.41542505382344)
(79.79192311073608, 0.706501553931617)
(67.2255524963769, 0.706501553931617)
(92.35829372509525, 0.706501553931617)
(-27.022227111316887, 0.706501553931617)
(-83.57089487593316, 0.706501553931617)
(-8.172671189778127, 0.706501553931617)
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:
$$x_{1} = -45.8717830328556$$
$$x_{2} = -77.2877095687536$$
$$x_{3} = -96.1372654902923$$
$$x_{4} = 10.6768847317606$$
$$x_{5} = 23.2432553461198$$
$$x_{6} = -20.7390418041373$$
$$x_{7} = 4.39369942458105$$
$$x_{8} = -1.88948588259854$$
$$x_{9} = 16.9600700389402$$
$$x_{10} = 54.6591818820177$$
$$x_{11} = 42.0928112676586$$
$$x_{12} = -52.1549683400352$$
$$x_{13} = -89.8540801831128$$
$$x_{14} = 35.809625960479$$
$$x_{15} = 60.9423671891973$$
$$x_{16} = 48.3759965748382$$
$$x_{17} = 86.0751084179157$$
$$x_{18} = -33.3054124184965$$
$$x_{19} = -39.5885977256761$$
$$x_{20} = -71.004524261574$$
$$x_{21} = 98.6414790322748$$
$$x_{22} = -64.7213389543944$$
$$x_{23} = 79.7919231107361$$
$$x_{24} = 67.2255524963769$$
$$x_{25} = 92.3582937250953$$
$$x_{26} = -27.0222271113169$$
$$x_{27} = -83.5708948759332$$
$$x_{28} = -8.17267118977813$$
Puntos máximos de la función:
$$x_{28} = -99.2788581438821$$
$$x_{28} = -86.712487529523$$
$$x_{28} = -17.5974491505475$$
$$x_{28} = 1.25210677099125$$
$$x_{28} = -74.1461169151638$$
$$x_{28} = -11.3142638433679$$
$$x_{28} = 32.6680333068892$$
$$x_{28} = 76.6503304571463$$
$$x_{28} = -61.5797463008046$$
$$x_{28} = 26.3848479997096$$
$$x_{28} = -36.4470050720863$$
$$x_{28} = 45.2344039212484$$
$$x_{28} = -23.8806344577271$$
$$x_{28} = 13.8184773853504$$
$$x_{28} = -30.1638197649067$$
$$x_{28} = -55.296560993625$$
$$x_{28} = 95.499886378685$$
$$x_{28} = -80.4293022223434$$
$$x_{28} = 51.5175892284279$$
$$x_{28} = 64.0839598427871$$
$$x_{28} = 89.2167010715055$$
$$x_{28} = 82.9335157643259$$
$$x_{28} = 57.8007745356075$$
$$x_{28} = -67.8629316079842$$
$$x_{28} = -42.7301903792658$$
$$x_{28} = -49.0133756864454$$
$$x_{28} = 38.9512186140688$$
$$x_{28} = -5.03107853618833$$
$$x_{28} = 20.10166269253$$
$$x_{28} = 7.53529207817084$$
$$x_{28} = 70.3671451499667$$
Decrece en los intervalos
$$\left[98.6414790322748, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -96.1372654902923\right]$$