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$$\left(- \frac{x}{4} + x\right) \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{4} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 48.7152107175577$$
$$x_{2} = 33.0170010333572$$
$$x_{3} = 76.9820093304187$$
$$x_{4} = -39.295350981473$$
$$x_{5} = 45.57503179559$$
$$x_{6} = 42.4350618814099$$
$$x_{7} = 51.855560729152$$
$$x_{8} = -36.1559664195367$$
$$x_{9} = -42.4350618814099$$
$$x_{10} = -2.02875783811043$$
$$x_{11} = -26.7409160147873$$
$$x_{12} = 0$$
$$x_{13} = 4.91318043943488$$
$$x_{14} = 83.2642147040886$$
$$x_{15} = -23.6042847729804$$
$$x_{16} = -92.687771772017$$
$$x_{17} = 64.4181717218392$$
$$x_{18} = 73.8409691490209$$
$$x_{19} = -95.8290108090195$$
$$x_{20} = -83.2642147040886$$
$$x_{21} = 29.8785865061074$$
$$x_{22} = -89.5465575382492$$
$$x_{23} = 17.3363779239834$$
$$x_{24} = -86.4053708116885$$
$$x_{25} = 11.085538406497$$
$$x_{26} = -4.91318043943488$$
$$x_{27} = 7.97866571241324$$
$$x_{28} = -29.8785865061074$$
$$x_{29} = -11.085538406497$$
$$x_{30} = -98.9702722883957$$
$$x_{31} = -17.3363779239834$$
$$x_{32} = 98.9702722883957$$
$$x_{33} = 54.9960525574964$$
$$x_{34} = 102.111554139654$$
$$x_{35} = 20.469167402741$$
$$x_{36} = 58.1366632448992$$
$$x_{37} = 39.295350981473$$
$$x_{38} = -64.4181717218392$$
$$x_{39} = 67.5590428388084$$
$$x_{40} = -51.855560729152$$
$$x_{41} = 86.4053708116885$$
$$x_{42} = 61.2773745335697$$
$$x_{43} = -80.1230928148503$$
$$x_{44} = 80.1230928148503$$
$$x_{45} = -76.9820093304187$$
$$x_{46} = -73.8409691490209$$
$$x_{47} = -67.5590428388084$$
$$x_{48} = -70.69997803861$$
$$x_{49} = 95.8290108090195$$
$$x_{50} = 92.687771772017$$
$$x_{51} = -48.7152107175577$$
$$x_{52} = -61.2773745335697$$
$$x_{53} = 2.02875783811043$$
$$x_{54} = 26.7409160147873$$
$$x_{55} = -33.0170010333572$$
$$x_{56} = 14.2074367251912$$
$$x_{57} = -58.1366632448992$$
$$x_{58} = 70.69997803861$$
$$x_{59} = 89.5465575382492$$
$$x_{60} = -7.97866571241324$$
$$x_{61} = -54.9960525574964$$
$$x_{62} = 36.1559664195367$$
$$x_{63} = -20.469167402741$$
$$x_{64} = -45.57503179559$$
$$x_{65} = -14.2074367251912$$
$$x_{66} = 23.6042847729804$$
Signos de extremos en los puntos:
(48.715210717557724, -36.528712669026)
(33.017001033357246, 24.751400798134)
(76.98200933041872, 57.7316363461978)
(-39.295350981472986, 29.4619747551688)
(45.57503179559002, 34.1730486270201)
(42.43506188140989, -31.8174630579443)
(51.85556072915197, 38.8844409376511)
(-36.15596641953672, -27.1066090291816)
(-42.43506188140989, -31.8174630579443)
(-2.028757838110434, 1.36477930586974)
(-26.74091601478731, 20.041678248523)
(0, 0)
(4.913180439434884, -3.6108524172842)
(83.26421470408864, 62.4436577797149)
(-23.604284772980407, -17.687347987225)
(-92.687771772017, -69.5117833410444)
(64.41817172183916, 48.3078085045315)
(73.8409691490209, -55.3756490786562)
(-95.82901080901948, 71.8678452063493)
(-83.26421470408864, 62.4436577797149)
(29.878586506107393, -22.3963996193901)
(-89.54655753824919, 67.1557307796639)
(17.33637792398336, -12.9807064558939)
(-86.40537081168854, -64.7996885367081)
(11.085538406497022, -8.2805310119475)
(-4.913180439434884, -3.6108524172842)
(7.978665712413241, 5.93754552869084)
(-29.878586506107393, -22.3963996193901)
(-11.085538406497022, -8.2805310119475)
(-98.9702722883957, -74.223915489839)
(-17.33637792398336, -12.9807064558939)
(98.9702722883957, -74.223915489839)
(54.99605255749639, -41.2402224372732)
(102.11155413965392, 76.5799934147373)
(20.46916740274095, 15.3335880436892)
(58.13666324489916, 43.5960485460643)
(39.295350981472986, 29.4619747551688)
(-64.41817172183916, 48.3078085045315)
(67.5590428388084, -50.6637323407294)
(-51.85556072915197, 38.8844409376511)
(86.40537081168854, -64.7996885367081)
(61.277374533569656, -45.9519124083574)
(-80.12309281485025, -60.0876398592444)
(80.12309281485025, -60.0876398592444)
(-76.98200933041872, 57.7316363461978)
(-73.8409691490209, -55.3756490786562)
(-67.5590428388084, -50.6637323407294)
(-70.69997803861, 53.0196802211948)
(95.82901080901948, 71.8678452063493)
(92.687771772017, -69.5117833410444)
(-48.715210717557724, -36.528712669026)
(-61.277374533569656, -45.9519124083574)
(2.028757838110434, 1.36477930586974)
(26.74091601478731, 20.041678248523)
(-33.017001033357246, 24.751400798134)
(14.207436725191188, 10.6292805853307)
(-58.13666324489916, 43.5960485460643)
(70.69997803861, 53.0196802211948)
(89.54655753824919, 67.1557307796639)
(-7.978665712413241, 5.93754552869084)
(-54.99605255749639, -41.2402224372732)
(36.15596641953672, -27.1066090291816)
(-20.46916740274095, 15.3335880436892)
(-45.57503179559002, 34.1730486270201)
(-14.207436725191188, 10.6292805853307)
(23.604284772980407, -17.687347987225)
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:
$$x_{1} = 48.7152107175577$$
$$x_{2} = 42.4350618814099$$
$$x_{3} = -36.1559664195367$$
$$x_{4} = -42.4350618814099$$
$$x_{5} = 0$$
$$x_{6} = 4.91318043943488$$
$$x_{7} = -23.6042847729804$$
$$x_{8} = -92.687771772017$$
$$x_{9} = 73.8409691490209$$
$$x_{10} = 29.8785865061074$$
$$x_{11} = 17.3363779239834$$
$$x_{12} = -86.4053708116885$$
$$x_{13} = 11.085538406497$$
$$x_{14} = -4.91318043943488$$
$$x_{15} = -29.8785865061074$$
$$x_{16} = -11.085538406497$$
$$x_{17} = -98.9702722883957$$
$$x_{18} = -17.3363779239834$$
$$x_{19} = 98.9702722883957$$
$$x_{20} = 54.9960525574964$$
$$x_{21} = 67.5590428388084$$
$$x_{22} = 86.4053708116885$$
$$x_{23} = 61.2773745335697$$
$$x_{24} = -80.1230928148503$$
$$x_{25} = 80.1230928148503$$
$$x_{26} = -73.8409691490209$$
$$x_{27} = -67.5590428388084$$
$$x_{28} = 92.687771772017$$
$$x_{29} = -48.7152107175577$$
$$x_{30} = -61.2773745335697$$
$$x_{31} = -54.9960525574964$$
$$x_{32} = 36.1559664195367$$
$$x_{33} = 23.6042847729804$$
Puntos máximos de la función:
$$x_{33} = 33.0170010333572$$
$$x_{33} = 76.9820093304187$$
$$x_{33} = -39.295350981473$$
$$x_{33} = 45.57503179559$$
$$x_{33} = 51.855560729152$$
$$x_{33} = -2.02875783811043$$
$$x_{33} = -26.7409160147873$$
$$x_{33} = 83.2642147040886$$
$$x_{33} = 64.4181717218392$$
$$x_{33} = -95.8290108090195$$
$$x_{33} = -83.2642147040886$$
$$x_{33} = -89.5465575382492$$
$$x_{33} = 7.97866571241324$$
$$x_{33} = 102.111554139654$$
$$x_{33} = 20.469167402741$$
$$x_{33} = 58.1366632448992$$
$$x_{33} = 39.295350981473$$
$$x_{33} = -64.4181717218392$$
$$x_{33} = -51.855560729152$$
$$x_{33} = -76.9820093304187$$
$$x_{33} = -70.69997803861$$
$$x_{33} = 95.8290108090195$$
$$x_{33} = 2.02875783811043$$
$$x_{33} = 26.7409160147873$$
$$x_{33} = -33.0170010333572$$
$$x_{33} = 14.2074367251912$$
$$x_{33} = -58.1366632448992$$
$$x_{33} = 70.69997803861$$
$$x_{33} = 89.5465575382492$$
$$x_{33} = -7.97866571241324$$
$$x_{33} = -20.469167402741$$
$$x_{33} = -45.57503179559$$
$$x_{33} = -14.2074367251912$$
Decrece en los intervalos
$$\left[98.9702722883957, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9702722883957\right]$$