Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d t} f{\left(t \right)} = $$
primera derivada$$- 3 \sin{\left(3 t \right)} - \frac{19 \sin{\left(\frac{19 t}{5} \right)}}{5} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$t_{1} = 46.2130785585998$$
$$t_{2} = -11.6625933562164$$
$$t_{3} = 80.3571482096613$$
$$t_{4} = 42.4028265792191$$
$$t_{5} = 51.8449530284747$$
$$t_{6} = 4.04536991173257$$
$$t_{7} = 8.30960459256762$$
$$t_{8} = -43.6089593406824$$
$$t_{9} = -66.3467835349603$$
$$t_{10} = 17.5252951378654$$
$$t_{11} = -13.8906313980325$$
$$t_{12} = -72.0508422820368$$
$$t_{13} = -64.6491849417123$$
$$t_{14} = -70.2302117471772$$
$$t_{15} = -5.58784584818021$$
$$t_{16} = -129.709076055324$$
$$t_{17} = 98.2931495194264$$
$$t_{18} = 70.2302117471772$$
$$t_{19} = -39.7255311284656$$
$$t_{20} = 61.9210418265488$$
$$t_{21} = 10.1201174197688$$
$$t_{22} = -46.2130785585998$$
$$t_{23} = 99.835625455874$$
$$t_{24} = 94.2477796076938$$
$$t_{25} = 31.4159265358979$$
$$t_{26} = 7.39835867538134$$
$$t_{27} = 36.1369897605258$$
$$t_{28} = -29.5985946659815$$
$$t_{29} = 72.0508422820368$$
$$t_{30} = 37.904900593606$$
$$t_{31} = 58.110789847168$$
$$t_{32} = 1.81733186991646$$
$$t_{33} = 43.0785198921143$$
$$t_{34} = -91.5399833739277$$
$$t_{35} = 0$$
$$t_{36} = 18.415759501715$$
$$t_{37} = 34.123722769664$$
$$t_{38} = 60.1240568380298$$
$$t_{39} = -51.8449530284747$$
$$t_{40} = -40.6349157461388$$
$$t_{41} = -23.1063219433303$$
$$t_{42} = -73.818753115117$$
$$t_{43} = 49.831686037613$$
$$t_{44} = -82.0547468029093$$
$$t_{45} = 13.8906313980325$$
$$t_{46} = -1.81733186991646$$
$$t_{47} = 84.127662187925$$
$$t_{48} = 92.4304477377773$$
$$t_{49} = 27.9009960727335$$
$$t_{50} = 12.1930328047845$$
$$t_{51} = 85.9381750151262$$
$$t_{52} = 82.0547468029093$$
$$t_{53} = -84.127662187925$$
$$t_{54} = -4.04536991173257$$
$$t_{55} = -75.8320201059788$$
$$t_{56} = -87.7588055499857$$
$$t_{57} = 66.3467835349603$$
$$t_{58} = 96.0651114776103$$
$$t_{59} = -61.9210418265488$$
$$t_{60} = -97.7627100708583$$
$$t_{61} = 68.4196989199761$$
$$t_{62} = 44.4160935700808$$
$$t_{63} = -99.835625455874$$
$$t_{64} = -25.8280806877177$$
$$t_{65} = 48.034701049094$$
$$t_{66} = -36.1369897605258$$
$$t_{67} = -48.034701049094$$
$$t_{68} = 25.8280806877177$$
$$t_{69} = 56.3428790140878$$
$$t_{70} = -22.196937325657$$
$$t_{71} = -77.6290050944977$$
$$t_{72} = 24.0175678605166$$
$$t_{73} = 90.2024096959612$$
$$t_{74} = -10.1201174197688$$
$$t_{75} = -85.9381750151262$$
$$t_{76} = -96.0651114776103$$
$$t_{77} = -27.9009960727335$$
$$t_{78} = -94.2477796076938$$
$$t_{79} = -37.904900593606$$
$$t_{80} = -19.7533331796815$$
$$t_{81} = -49.831686037613$$
$$t_{82} = 86.8494209323125$$
$$t_{83} = -24.0175678605166$$
$$t_{84} = -53.612863861555$$
$$t_{85} = -15.707963267949$$
$$t_{86} = 32.326737781145$$
$$t_{87} = 75.8320201059788$$
$$t_{88} = 22.196937325657$$
$$t_{89} = -67.5529162964237$$
$$t_{90} = -63.742664317043$$
$$t_{91} = -34.123722769664$$
$$t_{92} = -60.1240568380298$$
$$t_{93} = -58.110789847168$$
Signos de extremos en los puntos:
(46.2130785585998, 1.86685146053908)
(-11.662593356216396, 0.0353698741883327)
(80.35714820966129, -1.48631817948336)
(42.40282657921908, -0.58810826364559)
(51.84495302847472, -0.588108263645574)
(4.045369911732569, -0.0353698741883335)
(8.309604592567624, 1.96641492229032)
(-43.60895934068243, -0.267915527280037)
(-66.34678353496032, 0.267915527280091)
(17.525295137865424, -1.48631817948333)
(-13.890631398032507, -1.48631817948334)
(-72.05084228203678, -1.70486332547012)
(-64.64918494171232, 1.48631817948335)
(-70.23021174717721, -1.96641492229031)
(-5.587845848180208, -1.21939794095159)
(-129.7090760553243, -0.0353698741883438)
(98.29314951942636, -0.0353698741883313)
(70.23021174717721, -1.96641492229031)
(-39.725531128465555, 1.96641492229032)
(61.921041826548766, -1.86685146053908)
(10.120117419768759, 1.21939794095159)
(-46.2130785585998, 1.86685146053908)
(99.83562545587401, -1.21939794095161)
(94.2477796076938, 2)
(31.41592653589793, 2)
(7.398358675381342, -1.96641492229032)
(36.13698976052575, 0.588108263645588)
(-29.598594665981473, 1.48631817948333)
(72.05084228203678, -1.70486332547012)
(37.90490059360599, 1.70486332547013)
(58.11078984716804, 0.588108263645612)
(1.8173318699164585, 1.48631817948333)
(43.07851989211433, 0.0353698741883345)
(-91.53998337392774, -0.914915940591087)
(0, 2)
(18.41575950171503, 0.914915940591107)
(34.12372276966399, -0.914915940591114)
(60.1240568380298, -0.914915940591091)
(-51.84495302847472, -0.588108263645574)
(-40.634915746138844, -1.70486332547014)
(-23.106321943330308, 1.96641492229031)
(-73.81875311511702, -0.588108263645603)
(49.83168603761296, 0.914915940591105)
(-82.05474680290929, -0.267915527280083)
(13.890631398032507, -1.48631817948334)
(-1.8173318699164585, 1.48631817948333)
(84.12766218792504, 1.21939794095158)
(92.43044773777734, 1.48631817948333)
(27.900996072733466, 0.267915527280059)
(12.193032804784503, -0.267915527280059)
(85.93817501512618, 1.96641492229031)
(82.05474680290929, -0.267915527280083)
(-84.12766218792504, 1.21939794095158)
(-4.045369911732569, -0.0353698741883335)
(-75.83202010597877, 0.914915940591078)
(-87.75880554998574, 1.70486332547017)
(66.34678353496032, 0.267915527280091)
(96.06511147761026, 1.48631817948337)
(-61.921041826548766, -1.86685146053908)
(-97.76271007085826, 0.267915527280075)
(68.41969891997607, -1.21939794095157)
(44.41609357008084, 0.914915940591104)
(-99.83562545587401, -1.21939794095161)
(-25.828080687717726, -1.21939794095159)
(48.034701049094, 1.86685146053909)
(-36.13698976052575, 0.588108263645588)
(-48.034701049094, 1.86685146053909)
(25.828080687717726, -1.21939794095159)
(56.342879014087806, 1.70486332547014)
(-22.19693732565702, -1.70486332547014)
(-77.62900509449773, 1.86685146053907)
(24.01756786051659, -1.96641492229032)
(90.20240969596122, -0.0353698741883275)
(-10.120117419768759, 1.21939794095159)
(-85.93817501512618, 1.96641492229031)
(-96.06511147761026, 1.48631817948337)
(-27.900996072733466, 0.267915527280059)
(-94.2477796076938, 2)
(-37.90490059360599, 1.70486332547013)
(-19.753333179681537, 0.0353698741883297)
(-49.83168603761296, 0.914915940591105)
(86.84942093231246, -1.96641492229032)
(-24.01756786051659, -1.96641492229032)
(-53.612863861554956, -1.70486332547013)
(-15.707963267948966, -2)
(32.326737781145034, -1.86685146053909)
(75.83202010597877, 0.914915940591078)
(22.19693732565702, -1.70486332547014)
(-67.55291629642369, 0.588108263645561)
(-63.742664317042966, -1.86685146053909)
(-34.12372276966399, -0.914915940591114)
(-60.1240568380298, -0.914915940591091)
(-58.11078984716804, 0.588108263645612)
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:
$$t_{1} = 80.3571482096613$$
$$t_{2} = 42.4028265792191$$
$$t_{3} = 51.8449530284747$$
$$t_{4} = 4.04536991173257$$
$$t_{5} = -43.6089593406824$$
$$t_{6} = 17.5252951378654$$
$$t_{7} = -13.8906313980325$$
$$t_{8} = -72.0508422820368$$
$$t_{9} = -70.2302117471772$$
$$t_{10} = -5.58784584818021$$
$$t_{11} = -129.709076055324$$
$$t_{12} = 98.2931495194264$$
$$t_{13} = 70.2302117471772$$
$$t_{14} = 61.9210418265488$$
$$t_{15} = 99.835625455874$$
$$t_{16} = 7.39835867538134$$
$$t_{17} = 72.0508422820368$$
$$t_{18} = -91.5399833739277$$
$$t_{19} = 34.123722769664$$
$$t_{20} = 60.1240568380298$$
$$t_{21} = -51.8449530284747$$
$$t_{22} = -40.6349157461388$$
$$t_{23} = -73.818753115117$$
$$t_{24} = -82.0547468029093$$
$$t_{25} = 13.8906313980325$$
$$t_{26} = 12.1930328047845$$
$$t_{27} = 82.0547468029093$$
$$t_{28} = -4.04536991173257$$
$$t_{29} = -61.9210418265488$$
$$t_{30} = 68.4196989199761$$
$$t_{31} = -99.835625455874$$
$$t_{32} = -25.8280806877177$$
$$t_{33} = 25.8280806877177$$
$$t_{34} = -22.196937325657$$
$$t_{35} = 24.0175678605166$$
$$t_{36} = 90.2024096959612$$
$$t_{37} = 86.8494209323125$$
$$t_{38} = -24.0175678605166$$
$$t_{39} = -53.612863861555$$
$$t_{40} = -15.707963267949$$
$$t_{41} = 32.326737781145$$
$$t_{42} = 22.196937325657$$
$$t_{43} = -63.742664317043$$
$$t_{44} = -34.123722769664$$
$$t_{45} = -60.1240568380298$$
Puntos máximos de la función:
$$t_{45} = 46.2130785585998$$
$$t_{45} = -11.6625933562164$$
$$t_{45} = 8.30960459256762$$
$$t_{45} = -66.3467835349603$$
$$t_{45} = -64.6491849417123$$
$$t_{45} = -39.7255311284656$$
$$t_{45} = 10.1201174197688$$
$$t_{45} = -46.2130785585998$$
$$t_{45} = 94.2477796076938$$
$$t_{45} = 31.4159265358979$$
$$t_{45} = 36.1369897605258$$
$$t_{45} = -29.5985946659815$$
$$t_{45} = 37.904900593606$$
$$t_{45} = 58.110789847168$$
$$t_{45} = 1.81733186991646$$
$$t_{45} = 43.0785198921143$$
$$t_{45} = 0$$
$$t_{45} = 18.415759501715$$
$$t_{45} = -23.1063219433303$$
$$t_{45} = 49.831686037613$$
$$t_{45} = -1.81733186991646$$
$$t_{45} = 84.127662187925$$
$$t_{45} = 92.4304477377773$$
$$t_{45} = 27.9009960727335$$
$$t_{45} = 85.9381750151262$$
$$t_{45} = -84.127662187925$$
$$t_{45} = -75.8320201059788$$
$$t_{45} = -87.7588055499857$$
$$t_{45} = 66.3467835349603$$
$$t_{45} = 96.0651114776103$$
$$t_{45} = -97.7627100708583$$
$$t_{45} = 44.4160935700808$$
$$t_{45} = 48.034701049094$$
$$t_{45} = -36.1369897605258$$
$$t_{45} = -48.034701049094$$
$$t_{45} = 56.3428790140878$$
$$t_{45} = -77.6290050944977$$
$$t_{45} = -10.1201174197688$$
$$t_{45} = -85.9381750151262$$
$$t_{45} = -96.0651114776103$$
$$t_{45} = -27.9009960727335$$
$$t_{45} = -94.2477796076938$$
$$t_{45} = -37.904900593606$$
$$t_{45} = -19.7533331796815$$
$$t_{45} = -49.831686037613$$
$$t_{45} = 75.8320201059788$$
$$t_{45} = -67.5529162964237$$
$$t_{45} = -58.110789847168$$
Decrece en los intervalos
$$\left[99.835625455874, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -129.709076055324\right]$$