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$$x \cos{\left(7 x \right)} + \frac{\sin{\left(7 x \right)}}{7} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 50.0414908059181$$
$$x_{2} = 4.26836950087248$$
$$x_{3} = 10.0999968626586$$
$$x_{4} = -76.0716903870102$$
$$x_{5} = 36.1288803845179$$
$$x_{6} = -50.0414908059181$$
$$x_{7} = 0$$
$$x_{8} = 89.9844163742712$$
$$x_{9} = 74.2765010688857$$
$$x_{10} = -25.8067304541918$$
$$x_{11} = 100.306768896968$$
$$x_{12} = 37.4752569421903$$
$$x_{13} = -11.8948878148698$$
$$x_{14} = -24.0115937719656$$
$$x_{15} = 92.2284056087093$$
$$x_{16} = 54.0806509012589$$
$$x_{17} = -87.7404274220872$$
$$x_{18} = 40.168014138764$$
$$x_{19} = 67.9933411516917$$
$$x_{20} = -47.7975151995624$$
$$x_{21} = -37.9240494482814$$
$$x_{22} = 56.7734267059359$$
$$x_{23} = 96.2675868789753$$
$$x_{24} = -6.06215169734427$$
$$x_{25} = -54.5294467459485$$
$$x_{26} = 6.06215169734427$$
$$x_{27} = -91.7796077402351$$
$$x_{28} = -69.7885292329558$$
$$x_{29} = 47.3487202960603$$
$$x_{30} = -72.0325148757039$$
$$x_{31} = 18.1774801895039$$
$$x_{32} = 41.9631882059783$$
$$x_{33} = -77.8668800041013$$
$$x_{34} = -28.0506619473452$$
$$x_{35} = 70.2373263149385$$
$$x_{36} = 19.9725750897106$$
$$x_{37} = 28.0506619473452$$
$$x_{38} = -35.6800885389641$$
$$x_{39} = 14.1386103269137$$
$$x_{40} = -55.8758345793656$$
$$x_{41} = 98.0627788001522$$
$$x_{42} = 16.8311731368512$$
$$x_{43} = 26.2555158878374$$
$$x_{44} = -85.9452364784199$$
$$x_{45} = -95.8187889214026$$
$$x_{46} = -41.9631882059783$$
$$x_{47} = -19.9725750897106$$
$$x_{48} = 30.2946028113592$$
$$x_{49} = 84.1500457420413$$
$$x_{50} = 32.0897609286503$$
$$x_{51} = 0.289822548301491$$
$$x_{52} = -67.9933411516917$$
$$x_{53} = 52.2854680556737$$
$$x_{54} = 78.3156774526783$$
$$x_{55} = 62.1589829674753$$
$$x_{56} = -99.857970860865$$
$$x_{57} = -33.8849230387282$$
$$x_{58} = 85.9452364784199$$
$$x_{59} = -65.7493566429938$$
$$x_{60} = -3.82013085925533$$
$$x_{61} = -59.9150005114115$$
$$x_{62} = -98.0627788001522$$
$$x_{63} = -46.0023360593325$$
$$x_{64} = -15.9336435310187$$
$$x_{65} = -63.9541695536103$$
$$x_{66} = 58.119815230211$$
$$x_{67} = 76.0716903870102$$
$$x_{68} = -21.7676866303789$$
$$x_{69} = 72.0325148757039$$
$$x_{70} = 22.2164666429911$$
$$x_{71} = 44.2071582722925$$
$$x_{72} = 46.0023360593325$$
$$x_{73} = -61.7101864047208$$
$$x_{74} = -39.7192209289341$$
$$x_{75} = 94.0235971859024$$
$$x_{76} = -17.7287096567573$$
$$x_{77} = 8.30523760641417$$
$$x_{78} = -89.9844163742712$$
$$x_{79} = -94.0235971859024$$
$$x_{80} = 15.9336435310187$$
$$x_{81} = -1.13980938748761$$
$$x_{82} = -51.8366724844947$$
$$x_{83} = 88.1892251889095$$
$$x_{84} = -73.8277037886238$$
$$x_{85} = 63.9541695536103$$
$$x_{86} = -29.8458139903334$$
$$x_{87} = 33.8849230387282$$
$$x_{88} = -13.6898586870028$$
$$x_{89} = 2.02963381788446$$
$$x_{90} = 48.2463101783551$$
$$x_{91} = 66.1981534891744$$
$$x_{92} = -7.8565789367852$$
$$x_{93} = -43.7583640564846$$
$$x_{94} = -81.9060576338318$$
$$x_{95} = -2.02963381788446$$
$$x_{96} = -0.289822548301491$$
$$x_{97} = 24.0115937719656$$
$$x_{98} = -83.7012480918985$$
$$x_{99} = 80.1108674152685$$
Signos de extremos en los puntos:
(50.04149080591807, -7.14875527067737)
(4.268369500872485, -0.609425839983404)
(10.099996862658571, 1.44271238697129)
(-76.07169038701021, -10.8673651785235)
(36.128880384517934, 5.16122827889145)
(-50.04149080591807, -7.14875527067737)
(0, 0)
(89.98441637427116, 12.8549004251665)
(74.27650106888568, -10.610909098513)
(-25.80673045419181, -3.6866192941983)
(100.30676889696777, -14.3295238811969)
(37.47525694219035, -5.35356923666491)
(-11.894887814869808, 1.69914715046843)
(-24.0115937719656, -3.43016697406691)
(92.22840560870927, -13.1754707099512)
(54.080650901258934, 7.72578031708227)
(-87.74042742208724, -12.534330160543)
(40.16801413876397, -5.73825144373855)
(67.99334115169165, -9.71331301106694)
(-47.797515199562426, 6.82818595934055)
(-37.924049448281394, 5.41768291212835)
(56.77342670593591, 8.11046385333978)
(96.26758687897527, 13.7524972688694)
(-6.062151697344269, -0.865781307699165)
(-54.52944674594849, -7.78989423102338)
(6.062151697344269, -0.865781307699165)
(-91.77960774023505, 13.1113566514523)
(-69.78852923295581, -9.96976900272936)
(47.348720296060314, -6.76407211262846)
(-72.03251487570392, 10.2903390309626)
(18.17748018950395, 2.59670269386892)
(41.96318820597835, -5.99470643437687)
(-77.86688000410129, -11.1238212798889)
(-28.050661947345247, 4.0071854544471)
(70.23732631493846, 10.0338830050498)
(19.972575089710602, 2.85315202923658)
(28.050661947345247, 4.0071854544471)
(-35.68008853896408, -5.09711465061102)
(14.138610326913671, -2.01969838067589)
(-55.87583457936562, 7.9822359942291)
(98.0627788001522, 14.0089535348135)
(16.83117313685118, -2.40436670124825)
(26.255515887837383, 3.75073246445898)
(-85.9452364784199, -12.2778739644214)
(-95.81878892140257, -13.6883832040063)
(-41.96318820597835, -5.99470643437687)
(-19.972575089710602, 2.85315202923658)
(30.294602811359155, -4.3277522840869)
(84.15004574204129, -12.0214177831063)
(32.0897609286503, -4.58420613539779)
(0.2898225483014906, 0.0371368518604011)
(-67.99334115169165, -9.71331301106694)
(52.285468055673746, 7.4693246994023)
(78.31567745267834, 11.1879353083949)
(62.15898296747526, 8.87983125815889)
(-99.85797086086502, 14.2654098107245)
(-33.88492303872823, -4.84066027192193)
(85.9452364784199, -12.2778739644214)
(-65.74935664299379, 9.39274306383811)
(-3.82013085925533, 0.545351789075457)
(-59.91500051141154, -8.55926145754863)
(-98.0627788001522, 14.0089535348135)
(-46.002336059332485, 6.57173060633428)
(-15.933643531018715, -2.27614330836571)
(-63.9541695536103, 9.13628714302512)
(58.119815230210996, -8.30280566589329)
(76.07169038701021, -10.8673651785235)
(-21.767686630378858, 3.10960255336253)
(72.03251487570392, 10.2903390309626)
(22.216466642991147, -3.17371533637153)
(44.20715827229254, 6.31527534998677)
(46.002336059332485, 6.57173060633428)
(-61.71018640472079, -8.81571729292664)
(-39.71922092893406, 5.67413771800634)
(94.02359718590239, -13.4319269513249)
(-17.728709656757335, -2.53259058808252)
(8.305237606414167, 1.18628703526706)
(-89.98441637427116, 12.8549004251665)
(-94.02359718590239, -13.4319269513249)
(15.933643531018715, -2.27614330836571)
(-1.1398093874876059, 0.161565864726281)
(-51.83667248449475, -7.40521080499802)
(88.18922518890949, 12.5984442117809)
(-73.82770378862378, 10.5467950820297)
(63.9541695536103, 9.13628714302512)
(-29.845813990333443, 4.26363887185456)
(33.88492303872823, -4.84066027192193)
(-13.689858687002783, 1.95558762466256)
(2.0296338178844553, 0.289232124770904)
(48.246310178355095, -6.89229981143067)
(66.19815348917444, -9.45685704931534)
(-7.856578936785199, -1.12218292346322)
(-43.75836405648458, -6.25116155239689)
(-81.90605763383181, 11.700847578767)
(-2.0296338178844553, 0.289232124770904)
(-0.2898225483014906, 0.0371368518604011)
(24.0115937719656, -3.43016697406691)
(-83.70124809189848, 11.9573037402026)
(80.11086741526847, 11.444391434439)
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:
$$x_{1} = 50.0414908059181$$
$$x_{2} = 4.26836950087248$$
$$x_{3} = -76.0716903870102$$
$$x_{4} = -50.0414908059181$$
$$x_{5} = 0$$
$$x_{6} = 74.2765010688857$$
$$x_{7} = -25.8067304541918$$
$$x_{8} = 100.306768896968$$
$$x_{9} = 37.4752569421903$$
$$x_{10} = -24.0115937719656$$
$$x_{11} = 92.2284056087093$$
$$x_{12} = -87.7404274220872$$
$$x_{13} = 40.168014138764$$
$$x_{14} = 67.9933411516917$$
$$x_{15} = -6.06215169734427$$
$$x_{16} = -54.5294467459485$$
$$x_{17} = 6.06215169734427$$
$$x_{18} = -69.7885292329558$$
$$x_{19} = 47.3487202960603$$
$$x_{20} = 41.9631882059783$$
$$x_{21} = -77.8668800041013$$
$$x_{22} = -35.6800885389641$$
$$x_{23} = 14.1386103269137$$
$$x_{24} = 16.8311731368512$$
$$x_{25} = -85.9452364784199$$
$$x_{26} = -95.8187889214026$$
$$x_{27} = -41.9631882059783$$
$$x_{28} = 30.2946028113592$$
$$x_{29} = 84.1500457420413$$
$$x_{30} = 32.0897609286503$$
$$x_{31} = -67.9933411516917$$
$$x_{32} = -33.8849230387282$$
$$x_{33} = 85.9452364784199$$
$$x_{34} = -59.9150005114115$$
$$x_{35} = -15.9336435310187$$
$$x_{36} = 58.119815230211$$
$$x_{37} = 76.0716903870102$$
$$x_{38} = 22.2164666429911$$
$$x_{39} = -61.7101864047208$$
$$x_{40} = 94.0235971859024$$
$$x_{41} = -17.7287096567573$$
$$x_{42} = -94.0235971859024$$
$$x_{43} = 15.9336435310187$$
$$x_{44} = -51.8366724844947$$
$$x_{45} = 33.8849230387282$$
$$x_{46} = 48.2463101783551$$
$$x_{47} = 66.1981534891744$$
$$x_{48} = -7.8565789367852$$
$$x_{49} = -43.7583640564846$$
$$x_{50} = 24.0115937719656$$
Puntos máximos de la función:
$$x_{50} = 10.0999968626586$$
$$x_{50} = 36.1288803845179$$
$$x_{50} = 89.9844163742712$$
$$x_{50} = -11.8948878148698$$
$$x_{50} = 54.0806509012589$$
$$x_{50} = -47.7975151995624$$
$$x_{50} = -37.9240494482814$$
$$x_{50} = 56.7734267059359$$
$$x_{50} = 96.2675868789753$$
$$x_{50} = -91.7796077402351$$
$$x_{50} = -72.0325148757039$$
$$x_{50} = 18.1774801895039$$
$$x_{50} = -28.0506619473452$$
$$x_{50} = 70.2373263149385$$
$$x_{50} = 19.9725750897106$$
$$x_{50} = 28.0506619473452$$
$$x_{50} = -55.8758345793656$$
$$x_{50} = 98.0627788001522$$
$$x_{50} = 26.2555158878374$$
$$x_{50} = -19.9725750897106$$
$$x_{50} = 0.289822548301491$$
$$x_{50} = 52.2854680556737$$
$$x_{50} = 78.3156774526783$$
$$x_{50} = 62.1589829674753$$
$$x_{50} = -99.857970860865$$
$$x_{50} = -65.7493566429938$$
$$x_{50} = -3.82013085925533$$
$$x_{50} = -98.0627788001522$$
$$x_{50} = -46.0023360593325$$
$$x_{50} = -63.9541695536103$$
$$x_{50} = -21.7676866303789$$
$$x_{50} = 72.0325148757039$$
$$x_{50} = 44.2071582722925$$
$$x_{50} = 46.0023360593325$$
$$x_{50} = -39.7192209289341$$
$$x_{50} = 8.30523760641417$$
$$x_{50} = -89.9844163742712$$
$$x_{50} = -1.13980938748761$$
$$x_{50} = 88.1892251889095$$
$$x_{50} = -73.8277037886238$$
$$x_{50} = 63.9541695536103$$
$$x_{50} = -29.8458139903334$$
$$x_{50} = -13.6898586870028$$
$$x_{50} = 2.02963381788446$$
$$x_{50} = -81.9060576338318$$
$$x_{50} = -2.02963381788446$$
$$x_{50} = -0.289822548301491$$
$$x_{50} = -83.7012480918985$$
$$x_{50} = 80.1108674152685$$
Decrece en los intervalos
$$\left[100.306768896968, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.8187889214026\right]$$