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$$\frac{1}{7 x^{2}} \left(6 \sin{\left(6 x \right)} - 8 \sin{\left(8 x \right)}\right) - \frac{2 \left(- \cos{\left(6 x \right)} + \cos{\left(8 x \right)}\right)}{7 x^{3}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 15.994727371635$$
$$x_{2} = 100.242200315073$$
$$x_{3} = -5.99015122099848$$
$$x_{4} = -65.9734457253857$$
$$x_{5} = 67.9828838363912$$
$$x_{6} = -1.99039812592808$$
$$x_{7} = 70.2460452604027$$
$$x_{8} = -7.84888748827923$$
$$x_{9} = -51.8355071197721$$
$$x_{10} = -95.8181584782465$$
$$x_{11} = -75.398223686155$$
$$x_{12} = -72.5447419019908$$
$$x_{13} = 26.2628019920997$$
$$x_{14} = 52.2747454203449$$
$$x_{15} = 80.1101133557956$$
$$x_{16} = 84.1252244473817$$
$$x_{17} = 4.2638391503213$$
$$x_{18} = 41.971330885077$$
$$x_{19} = -49.976442371909$$
$$x_{20} = 8.28843467994347$$
$$x_{21} = -16.4029781723305$$
$$x_{22} = 50.5534280449663$$
$$x_{23} = 58.1187758508898$$
$$x_{24} = 21.9911485751286$$
$$x_{25} = 6.28318530717959$$
$$x_{26} = -61.6996391686419$$
$$x_{27} = -35.6879791078936$$
$$x_{28} = -87.9645943005142$$
$$x_{29} = 85.9541117260471$$
$$x_{30} = -17.7157487571002$$
$$x_{31} = -27.9848627582808$$
$$x_{32} = -19.9791411835855$$
$$x_{33} = -73.8268855538692$$
$$x_{34} = 92.2373284686778$$
$$x_{35} = 62.1339168426086$$
$$x_{36} = -9.71041808696057$$
$$x_{37} = 94.2477796076938$$
$$x_{38} = -55.850663112816$$
$$x_{39} = -67.9828838363912$$
$$x_{40} = -79.2366660769718$$
$$x_{41} = -97.3893722612836$$
$$x_{42} = 10.1183511081982$$
$$x_{43} = 89.9741748971613$$
$$x_{44} = -63.9628044779918$$
$$x_{45} = -45.9914565675393$$
$$x_{46} = -33.8590753222098$$
$$x_{47} = -91.8031021292985$$
$$x_{48} = 34.2682275706976$$
$$x_{49} = -39.708134960467$$
$$x_{50} = 96.2573889586344$$
$$x_{51} = -47.8204253527431$$
$$x_{52} = 23.9995189434553$$
$$x_{53} = 27.9848627582808$$
$$x_{54} = 45.9914565675393$$
$$x_{55} = 98.0863138651125$$
$$x_{56} = 56.259689006512$$
$$x_{57} = -53.6950526085793$$
$$x_{58} = 78.2509747301698$$
$$x_{59} = 32.1120741642892$$
$$x_{60} = 36.1272083392644$$
$$x_{61} = -23.9995189434553$$
$$x_{62} = -71.9677587831851$$
$$x_{63} = 88.2527727620244$$
$$x_{64} = -93.9589966811146$$
$$x_{65} = -31.7035476002572$$
$$x_{66} = -99.8332584305445$$
$$x_{67} = -81.6814089933346$$
$$x_{68} = 43.9822971502571$$
$$x_{69} = -29.8437899353152$$
$$x_{70} = -89.9741748971613$$
$$x_{71} = -11.8658652969711$$
$$x_{72} = 19.9791411835855$$
$$x_{73} = -69.8118235686108$$
$$x_{74} = -41.971330885077$$
$$x_{75} = 48.2546392943532$$
$$x_{76} = -21.9911485751286$$
$$x_{77} = -13.6950435141453$$
$$x_{78} = 14.1343373163651$$
$$x_{79} = 12.2756508084953$$
$$x_{80} = -85.9541117260471$$
$$x_{81} = 81.969563529006$$
$$x_{82} = 54.1037027097098$$
$$x_{83} = -25.8286013648995$$
$$x_{84} = 66.2615205765756$$
$$x_{85} = 74.2661184622797$$
$$x_{86} = -77.8420028672185$$
$$x_{87} = 1.99039812592808$$
$$x_{88} = -197.920337176157$$
$$x_{89} = 18.1501481883597$$
$$x_{90} = -43.9822971502571$$
$$x_{91} = 37.9868769247631$$
$$x_{92} = 59.9782916970685$$
$$x_{93} = 76.0950536850487$$
$$x_{94} = 30.283046563289$$
$$x_{95} = -3.82894549506131$$
$$x_{96} = 72.2566310325652$$
$$x_{97} = 40.1424351745792$$
$$x_{98} = 63.9628044779918$$
Signos de extremos en los puntos:
(15.994727371635012, -0.000286263195270719)
(100.24220031507328, -7.28892966359094e-6)
(-5.990151220998476, -0.00203968130085376)
(-65.97344572538566, 0)
(67.98288383639121, -5.58265424200495e-5)
(-1.9903981259280823, -0.0644923132070204)
(70.24604526040272, -5.22873188728933e-5)
(-7.848887488279229, 0.00463483137061056)
(-51.83550711977211, 0.000106333665995833)
(-95.81815847824647, 3.11196299223916e-5)
(-75.39822368615503, 0)
(-72.5447419019908, -1.39172034690113e-5)
(26.262801992099742, -0.000374055906494371)
(52.27474542034492, -9.44177117342441e-5)
(80.11011335579565, 4.45199385541152e-5)
(84.12522444738168, 2.55566275453332e-5)
(4.263839150321304, -0.0141609222981012)
(41.97133088507699, -0.000146463146741257)
(-49.976442371908995, -2.9324504299148e-5)
(8.28843467994347, -0.00375360678532376)
(-16.402978172330496, 0.000672127465025301)
(50.5534280449663, -2.86589467753649e-5)
(58.118775850889776, 8.45851056927603e-5)
(21.991148575128552, 0)
(6.283185307179586, 0)
(-61.69963916864185, -6.77756751233563e-5)
(-35.68797910789364, -0.000202575155806133)
(-87.96459430051421, 0)
(85.95411172604715, -3.49227132396698e-5)
(-17.715748757100172, -0.00082199620108839)
(-27.984862758280784, -9.35200126822674e-5)
(-19.9791411835855, -0.000646318823149579)
(-73.82688555386925, 5.24203206227868e-5)
(92.2373284686778, -3.03269102772175e-5)
(62.133916842608585, 4.68486002381012e-5)
(-9.710418086960566, -0.000776537685456993)
(94.2477796076938, 0)
(-55.850663112815994, 5.79824290293166e-5)
(-67.98288383639121, -5.58265424200495e-5)
(-79.2366660769718, 2.88073512956691e-5)
(-97.3893722612836, 0)
(10.118351108198247, 0.00176594632630256)
(89.97417489716135, -3.18717382776522e-5)
(-63.96280447799176, -6.30644216449555e-5)
(-45.99145656753926, -0.000121977901532855)
(-33.85907532220976, 0.000157758888274395)
(-91.80310212929847, 2.14605906976564e-5)
(34.26822757069757, -6.23695622348982e-5)
(-39.70813496046698, -0.00016363406880695)
(96.25738895863435, -2.78466919372928e-5)
(-47.82042535274311, 7.9090491024249e-5)
(23.99951894345535, -0.000447928487685137)
(27.984862758280784, -9.35200126822674e-5)
(45.99145656753926, -0.000121977901532855)
(98.08631386511246, 1.8799218668332e-5)
(56.259689006512, -2.31402318477698e-5)
(-53.69505260857925, -2.54034929003715e-5)
(78.25097473016984, -1.1961473102486e-5)
(32.11207416428923, 0.00017539033410453)
(36.127208339264435, 0.000218902055150067)
(-23.99951894345535, -0.000447928487685137)
(-71.96775878318509, -1.41412522386009e-5)
(88.25277276202439, -9.40389725636389e-6)
(-93.95899668111464, -8.29636853599508e-6)
(-31.703547600257153, -7.28682750574782e-5)
(-99.83325843054452, 1.81470566915843e-5)
(-81.68140899333463, 0)
(43.982297150257104, 0)
(-29.84378993531517, 0.000320777946666368)
(-89.97417489716135, -3.18717382776522e-5)
(-11.86586529697113, 0.00128423158080057)
(19.9791411835855, -0.000646318823149579)
(-69.81182356861079, 3.71105163139863e-5)
(-41.97133088507699, -0.000146463146741257)
(48.25463929435317, -0.000110804673011611)
(-21.991148575128552, 0)
(-13.69504351414526, -0.00137538675292933)
(14.134337316365114, 0.00142986196705477)
(12.275650808495291, -0.000485957553620244)
(-85.95411172604715, -3.49227132396698e-5)
(81.96956352900604, -1.09008183867297e-5)
(54.10370270970979, 6.17872320509756e-5)
(-25.828601364899537, 0.000271101468755938)
(66.26152057657558, -1.66817116127616e-5)
(74.26611846227972, -4.67798741671476e-5)
(-77.84200286721845, 2.98488513008963e-5)
(1.9903981259280823, -0.0644923132070204)
(-197.92033717615698, 0)
(18.150148188359747, 0.000548969544394231)
(-43.982297150257104, 0)
(37.98687692476311, -5.07563815300274e-5)
(59.978291697068514, -2.0359852139734e-5)
(76.0950536850487, 3.12350812888811e-5)
(30.283046563289012, -0.000281336274428901)
(-3.828945495061307, 0.0123045410761561)
(72.25663103256524, 0)
(40.142435174579205, 0.000112238207437807)
(63.96280447799176, -6.30644216449555e-5)
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:
$$x_{1} = 15.994727371635$$
$$x_{2} = 100.242200315073$$
$$x_{3} = -5.99015122099848$$
$$x_{4} = 67.9828838363912$$
$$x_{5} = -1.99039812592808$$
$$x_{6} = 70.2460452604027$$
$$x_{7} = -72.5447419019908$$
$$x_{8} = 26.2628019920997$$
$$x_{9} = 52.2747454203449$$
$$x_{10} = 4.2638391503213$$
$$x_{11} = 41.971330885077$$
$$x_{12} = -49.976442371909$$
$$x_{13} = 8.28843467994347$$
$$x_{14} = 50.5534280449663$$
$$x_{15} = -61.6996391686419$$
$$x_{16} = -35.6879791078936$$
$$x_{17} = 85.9541117260471$$
$$x_{18} = -17.7157487571002$$
$$x_{19} = -27.9848627582808$$
$$x_{20} = -19.9791411835855$$
$$x_{21} = 92.2373284686778$$
$$x_{22} = -9.71041808696057$$
$$x_{23} = -67.9828838363912$$
$$x_{24} = 89.9741748971613$$
$$x_{25} = -63.9628044779918$$
$$x_{26} = -45.9914565675393$$
$$x_{27} = 34.2682275706976$$
$$x_{28} = -39.708134960467$$
$$x_{29} = 96.2573889586344$$
$$x_{30} = 23.9995189434553$$
$$x_{31} = 27.9848627582808$$
$$x_{32} = 45.9914565675393$$
$$x_{33} = 56.259689006512$$
$$x_{34} = -53.6950526085793$$
$$x_{35} = 78.2509747301698$$
$$x_{36} = -23.9995189434553$$
$$x_{37} = -71.9677587831851$$
$$x_{38} = 88.2527727620244$$
$$x_{39} = -93.9589966811146$$
$$x_{40} = -31.7035476002572$$
$$x_{41} = -89.9741748971613$$
$$x_{42} = 19.9791411835855$$
$$x_{43} = -41.971330885077$$
$$x_{44} = 48.2546392943532$$
$$x_{45} = -13.6950435141453$$
$$x_{46} = 12.2756508084953$$
$$x_{47} = -85.9541117260471$$
$$x_{48} = 81.969563529006$$
$$x_{49} = 66.2615205765756$$
$$x_{50} = 74.2661184622797$$
$$x_{51} = 1.99039812592808$$
$$x_{52} = 37.9868769247631$$
$$x_{53} = 59.9782916970685$$
$$x_{54} = 30.283046563289$$
$$x_{55} = 63.9628044779918$$
Puntos máximos de la función:
$$x_{55} = -65.9734457253857$$
$$x_{55} = -7.84888748827923$$
$$x_{55} = -51.8355071197721$$
$$x_{55} = -95.8181584782465$$
$$x_{55} = -75.398223686155$$
$$x_{55} = 80.1101133557956$$
$$x_{55} = 84.1252244473817$$
$$x_{55} = -16.4029781723305$$
$$x_{55} = 58.1187758508898$$
$$x_{55} = 21.9911485751286$$
$$x_{55} = 6.28318530717959$$
$$x_{55} = -87.9645943005142$$
$$x_{55} = -73.8268855538692$$
$$x_{55} = 62.1339168426086$$
$$x_{55} = 94.2477796076938$$
$$x_{55} = -55.850663112816$$
$$x_{55} = -79.2366660769718$$
$$x_{55} = -97.3893722612836$$
$$x_{55} = 10.1183511081982$$
$$x_{55} = -33.8590753222098$$
$$x_{55} = -91.8031021292985$$
$$x_{55} = -47.8204253527431$$
$$x_{55} = 98.0863138651125$$
$$x_{55} = 32.1120741642892$$
$$x_{55} = 36.1272083392644$$
$$x_{55} = -99.8332584305445$$
$$x_{55} = -81.6814089933346$$
$$x_{55} = 43.9822971502571$$
$$x_{55} = -29.8437899353152$$
$$x_{55} = -11.8658652969711$$
$$x_{55} = -69.8118235686108$$
$$x_{55} = -21.9911485751286$$
$$x_{55} = 14.1343373163651$$
$$x_{55} = 54.1037027097098$$
$$x_{55} = -25.8286013648995$$
$$x_{55} = -77.8420028672185$$
$$x_{55} = -197.920337176157$$
$$x_{55} = 18.1501481883597$$
$$x_{55} = -43.9822971502571$$
$$x_{55} = 76.0950536850487$$
$$x_{55} = -3.82894549506131$$
$$x_{55} = 72.2566310325652$$
$$x_{55} = 40.1424351745792$$
Decrece en los intervalos
$$\left[100.242200315073, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -93.9589966811146\right]$$