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$$\frac{8 t e^{2 t}}{3} - \frac{16 e^{2 t}}{9} - \frac{50 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{9} + \frac{7 \cos{\left(\sqrt{2} t \right)}}{9} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$t_{1} = -82.1235616873772$$
$$t_{2} = -66.5734714038229$$
$$t_{3} = -4.37330165985759$$
$$t_{4} = -91.0093275636939$$
$$t_{5} = -53.2448225893478$$
$$t_{6} = -51.0233811202686$$
$$t_{7} = -57.6877055275062$$
$$t_{8} = -55.466264058427$$
$$t_{9} = -95.4522105018523$$
$$t_{10} = -17.7017590840809$$
$$t_{11} = -75.4592372801396$$
$$t_{12} = -86.5664446255356$$
$$t_{13} = -46.5804981821103$$
$$t_{14} = -24.3660834913184$$
$$t_{15} = -93.2307690327731$$
$$t_{16} = -0.0830671258322571$$
$$t_{17} = 1.03338755252582$$
$$t_{18} = -26.5875249603976$$
$$t_{19} = -84.3450031564564$$
$$t_{20} = -59.9091469965854$$
$$t_{21} = -19.9232005531601$$
$$t_{22} = -37.6947323057935$$
$$t_{23} = -44.3590567130311$$
$$t_{24} = -77.6806787492188$$
$$t_{25} = -31.030407898556$$
$$t_{26} = -13.2588761459326$$
$$t_{27} = -2.14242641424928$$
$$t_{28} = -73.2377958110605$$
$$t_{29} = -42.1376152439519$$
$$t_{30} = -88.7878860946147$$
$$t_{31} = -15.4803176150016$$
$$t_{32} = -11.0374346761198$$
$$t_{33} = -97.6736519709315$$
$$t_{34} = -33.2518493676352$$
$$t_{35} = -64.3520299347437$$
$$t_{36} = -62.1305884656645$$
$$t_{37} = -22.1446420222392$$
$$t_{38} = -6.59454849416224$$
$$t_{39} = -35.4732908367143$$
$$t_{40} = -79.902120218298$$
$$t_{41} = -71.0163543419813$$
$$t_{42} = -39.9161737748727$$
$$t_{43} = -99.8950934400107$$
Signos de extremos en los puntos:
/ ___\ ___ / ___\
50*cos\82.1235616873772*\/ 2 / 7*\/ 2 *sin\82.1235616873772*\/ 2 /
(-82.12356168737719, -5.17503676202054e-70 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\66.5734714038229*\/ 2 / 7*\/ 2 *sin\66.5734714038229*\/ 2 /
(-66.57347140382291, -1.35145711943326e-56 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\4.37330165985759*\/ 2 / 7*\/ 2 *sin\4.37330165985759*\/ 2 /
(-4.373301659857594, -0.00117447685096334 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\91.0093275636939*\/ 2 / 7*\/ 2 *sin\91.0093275636939*\/ 2 /
(-91.00932756369393, -1.09612237246476e-77 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\53.2448225893478*\/ 2 / 7*\/ 2 *sin\53.2448225893478*\/ 2 /
(-53.24482258934781, -4.09981319144366e-45 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\51.0233811202686*\/ 2 / 7*\/ 2 *sin\51.0233811202686*\/ 2 /
(-51.02338112026863, -3.34334115220086e-43 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\57.6877055275062*\/ 2 / 7*\/ 2 *sin\57.6877055275062*\/ 2 /
(-57.68770552750618, -6.13497926864371e-49 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\55.466264058427*\/ 2 / 7*\/ 2 *sin\55.466264058427*\/ 2 /
(-55.466264058426994, -5.01906648510858e-47 + ----------------------------- - ----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\95.4522105018523*\/ 2 / 7*\/ 2 *sin\95.4522105018523*\/ 2 /
(-95.45221050185229, -1.58951309411117e-81 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\17.7017590840809*\/ 2 / 7*\/ 2 *sin\17.7017590840809*\/ 2 /
(-17.701759084080884, -1.05955089679186e-14 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\75.4592372801396*\/ 2 / 7*\/ 2 *sin\75.4592372801396*\/ 2 /
(-75.45923728013965, -2.92586058645026e-64 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\86.5664446255356*\/ 2 / 7*\/ 2 *sin\86.5664446255356*\/ 2 /
(-86.56644462553555, -7.54125853732662e-74 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\46.5804981821103*\/ 2 / 7*\/ 2 *sin\46.5804981821103*\/ 2 /
(-46.58049818211026, -2.2109538423037e-39 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\24.3660834913184*\/ 2 / 7*\/ 2 *sin\24.3660834913184*\/ 2 /
(-24.36608349131843, -2.33305577035163e-20 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\93.2307690327731*\/ 2 / 7*\/ 2 *sin\93.2307690327731*\/ 2 /
(-93.23076903277311, -1.32032809432067e-79 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\0.0830671258322571*\/ 2 / 7*\/ 2 *sin\0.0830671258322571*\/ 2 /
(-0.08306712583225709, -1.4112535943714 + -------------------------------- - -------------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\1.03338755252582*\/ 2 / 7*\/ 2 *sin\1.03338755252582*\/ 2 /
(1.0333875525258183, -1.4037502853051 + ------------------------------ + -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\26.5875249603976*\/ 2 / 7*\/ 2 *sin\26.5875249603976*\/ 2 /
(-26.587524960397616, -2.98288536671326e-22 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\84.3450031564564*\/ 2 / 7*\/ 2 *sin\84.3450031564564*\/ 2 /
(-84.34500315645637, -6.24921159442002e-72 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\59.9091469965854*\/ 2 / 7*\/ 2 *sin\59.9091469965854*\/ 2 /
(-59.90914699658536, -7.48831466898766e-51 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\19.9232005531601*\/ 2 / 7*\/ 2 *sin\19.9232005531601*\/ 2 /
(-19.923200553160065, -1.39296581317188e-16 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\37.6947323057935*\/ 2 / 7*\/ 2 *sin\37.6947323057935*\/ 2 /
(-37.69473230579353, -9.40218284899873e-32 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\44.3590567130311*\/ 2 / 7*\/ 2 *sin\44.3590567130311*\/ 2 /
(-44.35905671303108, -1.79229084338781e-37 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\77.6806787492188*\/ 2 / 7*\/ 2 *sin\77.6806787492188*\/ 2 /
(-77.68067874921883, -3.54116000398269e-66 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\31.030407898556*\/ 2 / 7*\/ 2 *sin\31.030407898556*\/ 2 /
(-31.030407898555982, -4.78724117123825e-26 + ----------------------------- - ----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\13.2588761459326*\/ 2 / 7*\/ 2 *sin\13.2588761459326*\/ 2 /
(-13.258876145932641, -5.85541131734793e-11 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\2.14242641424928*\/ 2 / 7*\/ 2 *sin\2.14242641424928*\/ 2 /
(-2.1424264142492846, -0.0607798719738458 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\73.2377958110605*\/ 2 / 7*\/ 2 *sin\73.2377958110605*\/ 2 /
(-73.23779581106047, -2.41544161934843e-62 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\42.1376152439519*\/ 2 / 7*\/ 2 *sin\42.1376152439519*\/ 2 /
(-42.1376152439519, -1.44944591939558e-35 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\88.7878860946147*\/ 2 / 7*\/ 2 *sin\88.7878860946147*\/ 2 /
(-88.78788609461475, -9.09460630662491e-76 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\15.4803176150016*\/ 2 / 7*\/ 2 *sin\15.4803176150016*\/ 2 /
(-15.480317615001562, -7.94769641190152e-13 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\11.0374346761198*\/ 2 / 7*\/ 2 *sin\11.0374346761198*\/ 2 /
(-11.037434676119835, -4.21163371416618e-9 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\97.6736519709315*\/ 2 / 7*\/ 2 *sin\97.6736519709315*\/ 2 /
(-97.67365197093147, -1.91256725857317e-83 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\33.2518493676352*\/ 2 / 7*\/ 2 *sin\33.2518493676352*\/ 2 /
(-33.25184936763517, -6.01923766389167e-28 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\64.3520299347437*\/ 2 / 7*\/ 2 *sin\64.3520299347437*\/ 2 /
(-64.35202993474373, -1.11132474852584e-54 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\62.1305884656645*\/ 2 / 7*\/ 2 *sin\62.1305884656645*\/ 2 /
(-62.130588465664545, -9.12809513816115e-53 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\22.1446420222392*\/ 2 / 7*\/ 2 *sin\22.1446420222392*\/ 2 /
(-22.14464202223925, -1.81098029144111e-18 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\6.59454849416224*\/ 2 / 7*\/ 2 *sin\6.59454849416224*\/ 2 /
(-6.594548494162237, -1.93604932036221e-5 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\35.4732908367143*\/ 2 / 7*\/ 2 *sin\35.4732908367143*\/ 2 /
(-35.47329083671435, -7.53676141209859e-30 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\79.902120218298*\/ 2 / 7*\/ 2 *sin\79.902120218298*\/ 2 /
(-79.90212021829801, -4.28245299264831e-68 + ----------------------------- - ----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\71.0163543419813*\/ 2 / 7*\/ 2 *sin\71.0163543419813*\/ 2 /
(-71.01635434198128, -1.9922881858096e-60 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\39.9161737748727*\/ 2 / 7*\/ 2 *sin\39.9161737748727*\/ 2 /
(-39.91617377487271, -1.1690987096601e-33 + ------------------------------ - -----------------------------------)
9 18
/ ___\ ___ / ___\
50*cos\99.8950934400107*\/ 2 / 7*\/ 2 *sin\99.8950934400107*\/ 2 /
(-99.89509344001065, -2.30011682032021e-85 + ------------------------------ - -----------------------------------)
9 18
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:
$$t_{1} = -82.1235616873772$$
$$t_{2} = -91.0093275636939$$
$$t_{3} = -51.0233811202686$$
$$t_{4} = -55.466264058427$$
$$t_{5} = -95.4522105018523$$
$$t_{6} = -86.5664446255356$$
$$t_{7} = -46.5804981821103$$
$$t_{8} = -24.3660834913184$$
$$t_{9} = 1.03338755252582$$
$$t_{10} = -59.9091469965854$$
$$t_{11} = -19.9232005531601$$
$$t_{12} = -37.6947323057935$$
$$t_{13} = -77.6806787492188$$
$$t_{14} = -2.14242641424928$$
$$t_{15} = -73.2377958110605$$
$$t_{16} = -42.1376152439519$$
$$t_{17} = -15.4803176150016$$
$$t_{18} = -11.0374346761198$$
$$t_{19} = -33.2518493676352$$
$$t_{20} = -64.3520299347437$$
$$t_{21} = -6.59454849416224$$
$$t_{22} = -99.8950934400107$$
Puntos máximos de la función:
$$t_{22} = -66.5734714038229$$
$$t_{22} = -4.37330165985759$$
$$t_{22} = -53.2448225893478$$
$$t_{22} = -57.6877055275062$$
$$t_{22} = -17.7017590840809$$
$$t_{22} = -75.4592372801396$$
$$t_{22} = -93.2307690327731$$
$$t_{22} = -0.0830671258322571$$
$$t_{22} = -26.5875249603976$$
$$t_{22} = -84.3450031564564$$
$$t_{22} = -44.3590567130311$$
$$t_{22} = -31.030407898556$$
$$t_{22} = -13.2588761459326$$
$$t_{22} = -88.7878860946147$$
$$t_{22} = -97.6736519709315$$
$$t_{22} = -62.1305884656645$$
$$t_{22} = -22.1446420222392$$
$$t_{22} = -35.4732908367143$$
$$t_{22} = -79.902120218298$$
$$t_{22} = -71.0163543419813$$
$$t_{22} = -39.9161737748727$$
Decrece en los intervalos
$$\left[1.03338755252582, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.8950934400107\right]$$