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Trazado el gráfico de la función/ -exp(x)/9+x*exp(x)/3+cos(x*sqrt(5))+sin(x*sqrt(5))

Gráfico de la función y = -exp(x)/9+x*exp(x)/3+cos(x*sqrt(5))+sin(x*sqrt(5))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         x       x                              
       -e     x*e       /    ___\      /    ___\
f(x) = ---- + ---- + cos\x*\/ 5 / + sin\x*\/ 5 /
        9      3
f(x)=((xex3+(−1)ex9)+cos⁡(5x))+sin⁡(5x)f{\left(x \right)} = \left(\left(\frac{x e^{x}}{3} + \frac{\left(-1\right) e^{x}}{9}\right) + \cos{\left(\sqrt{5} x \right)}\right) + \sin{\left(\sqrt{5} x \right)} 
f = (x*exp(x))/3 + (-exp(x))/9 + cos(sqrt(5)*x) + sin(sqrt(5)*x)
Gráfico de la función
02468-8-6-4-2-1010-100000100000
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
((xex3+(−1)ex9)+cos⁡(5x))+sin⁡(5x)=0\left(\left(\frac{x e^{x}}{3} + \frac{\left(-1\right) e^{x}}{9}\right) + \cos{\left(\sqrt{5} x \right)}\right) + \sin{\left(\sqrt{5} x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−15.8058333773152x_{1} = -15.8058333773152
x2=−29.8554626069234x_{2} = -29.8554626069234
x3=−0.3016359558908x_{3} = -0.3016359558908
x4=−87.458943401457x_{4} = -87.458943401457
x5=−62.1696103697104x_{5} = -62.1696103697104
x6=−24.2356108221678x_{6} = -24.2356108221678
x7=−76.2192398317919x_{7} = -76.2192398317919
x8=−39.6902032303801x_{8} = -39.6902032303801
x9=−72.0043509931674x_{9} = -72.0043509931674
x10=−38.285240284172x_{10} = -38.285240284172
x11=−3.14537716752517x_{11} = -3.14537716752517
x12=−49.5249438538371x_{12} = -49.5249438538371
x13=−5.9693945387805x_{13} = -5.9693945387805
x14=−66.3844992083349x_{14} = -66.3844992083349
x15=−63.5745733159186x_{15} = -63.5745733159186
x16=−10.1860231559657x_{16} = -10.1860231559657
x17=−1.79354582985988x_{17} = -1.79354582985988
x18=−86.0539804552489x_{18} = -86.0539804552489
x19=−34.0703514455475x_{19} = -34.0703514455475
x20=−7.37656400908112x_{20} = -7.37656400908112
x21=−21.4256849308104x_{21} = -21.4256849308104
x22=−59.3596844772941x_{22} = -59.3596844772941
x23=−73.4093139393756x_{23} = -73.4093139393756
x24=−35.4753143917557x_{24} = -35.4753143917557
x25=−100.10360991733x_{25} = -100.10360991733
x26=−43.9050920690045x_{26} = -43.9050920690045
x27=−95.8887210787059x_{27} = -95.8887210787059
x28=−48.119980907629x_{28} = -48.119980907629
x29=−80.4341286704163x_{29} = -80.4341286704163
x30=−25.6405737682786x_{30} = -25.6405737682786
x31=−57.954721531086x_{31} = -57.954721531086
x32=−91.6738322400815x_{32} = -91.6738322400815
x33=−45.3100550152127x_{33} = -45.3100550152127
x34=−107.128424648371x_{34} = -107.128424648371
x35=−67.789462154543x_{35} = -67.789462154543
x36=−28.4504996607136x_{36} = -28.4504996607136
x37=−14.4008693336909x_{37} = -14.4008693336909
x38=−77.624202778x_{38} = -77.624202778
x39=−20.0207219791345x_{39} = -20.0207219791345
x40=−81.8390916166245x_{40} = -81.8390916166245
x41=−53.7398326924616x_{41} = -53.7398326924616
x42=−90.2688692938733x_{42} = -90.2688692938733
x43=−55.1447956386697x_{43} = -55.1447956386697
x44=−11.5909326801642x_{44} = -11.5909326801642
x45=−52.3348697462534x_{45} = -52.3348697462534
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (-exp(x))/9 + (x*exp(x))/3 + cos(x*sqrt(5)) + sin(x*sqrt(5)).
sin⁡(05)+(((−1)e09+0e03)+cos⁡(05))\sin{\left(0 \sqrt{5} \right)} + \left(\left(\frac{\left(-1\right) e^{0}}{9} + \frac{0 e^{0}}{3}\right) + \cos{\left(0 \sqrt{5} \right)}\right)
Resultado:
f(0)=89f{\left(0 \right)} = \frac{8}{9}
Punto:
(0, 8/9)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
xex3+2ex9−5sin⁡(5x)+5cos⁡(5x)=0\frac{x e^{x}}{3} + \frac{2 e^{x}}{9} - \sqrt{5} \sin{\left(\sqrt{5} x \right)} + \sqrt{5} \cos{\left(\sqrt{5} x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−71.3018695200634x_{1} = -71.3018695200634
x2=−64.2770547890226x_{2} = -64.2770547890226
x3=−75.5167583586878x_{3} = -75.5167583586878
x4=−74.1117954124797x_{4} = -74.1117954124797
x5=−78.3266842511041x_{5} = -78.3266842511041
x6=−17.9132775506499x_{6} = -17.9132775506499
x7=−65.6820177352308x_{7} = -65.6820177352308
x8=−19.3182405139455x_{8} = -19.3182405139455
x9=−3.86047764079769x_{9} = -3.86047764079769
x10=−83.9465360359367x_{10} = -83.9465360359367
x11=−33.3678699724435x_{11} = -33.3678699724435
x12=−60.0621659503982x_{12} = -60.0621659503982
x13=−8.07864528768309x_{13} = -8.07864528768309
x14=−69.8969065738552x_{14} = -69.8969065738552
x15=−237.087497172625x_{15} = -237.087497172625
x16=−30.5579440800272x_{16} = -30.5579440800272
x17=−79.7316471973122x_{17} = -79.7316471973122
x18=−61.4671288966064x_{18} = -61.4671288966064
x19=−12.2934232680916x_{19} = -12.2934232680916
x20=−47.4174994345249x_{20} = -47.4174994345249
x21=0.431345129440401x_{21} = 0.431345129440401
x22=−54.4423141655656x_{22} = -54.4423141655656
x23=−51.6323882731493x_{23} = -51.6323882731493
x24=−46.0125364883168x_{24} = -46.0125364883168
x25=−99.4011284442263x_{25} = -99.4011284442263
x26=−9.48346825780183x_{26} = -9.48346825780183
x27=−22.1281664030266x_{27} = -22.1281664030266
x28=−31.9629070262353x_{28} = -31.9629070262353
x29=−50.2274253269412x_{29} = -50.2274253269412
x30=−26.3430552413983x_{30} = -26.3430552413983
x31=−13.6983894161836x_{31} = -13.6983894161836
x32=−27.748018187612x_{32} = -27.748018187612
x33=−55.8472771117738x_{33} = -55.8472771117738
x34=−41.7976476496923x_{34} = -41.7976476496923
x35=−88.1614248745611x_{35} = -88.1614248745611
x36=−23.5331293489215x_{36} = -23.5331293489215
x37=1.33771825628215x_{37} = 1.33771825628215
x38=−93.7812766593937x_{38} = -93.7812766593937
x39=−37.5827588110679x_{39} = -37.5827588110679
x40=−36.1777958648597x_{40} = -36.1777958648597
x41=−89.5663878207693x_{41} = -89.5663878207693
x42=−97.9961654980181x_{42} = -97.9961654980181
x43=−40.3926847034842x_{43} = -40.3926847034842
x44=−92.3763137131856x_{44} = -92.3763137131856
x45=−34.7728329186516x_{45} = -34.7728329186516
x46=−85.3514989821448x_{46} = -85.3514989821448
Signos de extremos en los puntos:
                                                /                   ___\      /                   ___\ 
(-71.30186952006338, -2.58224091641026e-30 - sin\71.3018695200634*\/ 5 / + cos\71.3018695200634*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-64.27705478902264, -2.61824864423254e-27 - sin\64.2770547890226*\/ 5 / + cos\64.2770547890226*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-75.51675835868781, -4.0394609025538e-32 - sin\75.5167583586878*\/ 5 / + cos\75.5167583586878*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-74.11179541247967, -1.6157391157327e-31 - sin\74.1117954124797*\/ 5 / + cos\74.1117954124797*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-78.3266842511041, -2.52223779051979e-33 - sin\78.3266842511041*\/ 5 / + cos\78.3266842511041*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-17.91327755064993, -1.01023721482142e-7 - sin\17.9132775506499*\/ 5 / + cos\17.9132775506499*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-65.68201773523079, -6.56426068548037e-28 - sin\65.6820177352308*\/ 5 / + cos\65.6820177352308*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-19.31824051394547, -2.66975142934792e-8 - sin\19.3182405139455*\/ 5 / + cos\19.3182405139455*\/ 5 /)

                                             /                   ___\      /                   ___\ 
(-3.860477640797693, -0.029437671875995 - sin\3.86047764079769*\/ 5 / + cos\3.86047764079769*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-83.94653603593667, -9.79682214483807e-36 - sin\83.9465360359367*\/ 5 / + cos\83.9465360359367*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-33.36786997244346, -3.6227832496974e-14 - sin\33.3678699724435*\/ 5 / + cos\33.3678699724435*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-60.06216595039821, -1.65659397594601e-25 - sin\60.0621659503982*\/ 5 / + cos\60.0621659503982*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-8.078645287683095, -0.000869492491177853 - sin\8.07864528768309*\/ 5 / + cos\8.07864528768309*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-69.89690657385523, -1.03172052419022e-29 - sin\69.8969065738552*\/ 5 / + cos\69.8969065738552*\/ 5 /)

                                                  /                   ___\      /                   ___\ 
(-237.08749717262452, -8.56260312031915e-102 - sin\237.087497172625*\/ 5 / + cos\237.087497172625*\/ 5 /)

                                                 /                   ___\      /                   ___\ 
(-30.557944080027234, -5.51528704787994e-13 - sin\30.5579440800272*\/ 5 / + cos\30.5579440800272*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-79.73164719731224, -6.2995124242245e-34 - sin\79.7316471973122*\/ 5 / + cos\79.7316471973122*\/ 5 /)

                                                 /                   ___\      /                   ___\ 
(-61.467128896606354, -4.15944676586777e-26 - sin\61.4671288966064*\/ 5 / + cos\61.4671288966064*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-12.293423268091594, -1.92843349060251e-5 - sin\12.2934232680916*\/ 5 / + cos\12.2934232680916*\/ 5 /)

                                                 /                   ___\      /                   ___\ 
(-47.417499434524906, -4.06163923108672e-20 - sin\47.4174994345249*\/ 5 / + cos\47.4174994345249*\/ 5 /)

                                              /                    ___\      /                    ___\ 
(0.43134512944040104, 0.0502907257128059 + cos\0.431345129440401*\/ 5 / + sin\0.431345129440401*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-54.44231416556563, -4.14447611555443e-23 - sin\54.4423141655656*\/ 5 / + cos\54.4423141655656*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-51.63238827314934, -6.53031894671202e-22 - sin\51.6323882731493*\/ 5 / + cos\51.6323882731493*\/ 5 /)

                                                 /                   ___\      /                   ___\ 
(-46.012536488316755, -1.60656790800255e-19 - sin\46.0125364883168*\/ 5 / + cos\46.0125364883168*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-99.40112844422627, -2.25093220745963e-42 - sin\99.4011284442263*\/ 5 / + cos\99.4011284442263*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-9.483468257801832, -0.000249018048371451 - sin\9.48346825780183*\/ 5 / + cos\9.48346825780183*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-22.12816640302655, -1.83728670414608e-9 - sin\22.1281664030266*\/ 5 / + cos\22.1281664030266*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-31.962907026235285, -1.414870093174e-13 - sin\31.9629070262353*\/ 5 / + cos\31.9629070262353*\/ 5 /)

                                                 /                   ___\      /                   ___\ 
(-50.227425326941194, -2.58939716071696e-21 - sin\50.2274253269412*\/ 5 / + cos\50.2274253269412*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-26.343055241398336, -3.2237566595419e-11 - sin\26.3430552413983*\/ 5 / + cos\26.3430552413983*\/ 5 /)

                                              /                   ___\      /                   ___\ 
(-13.69838941618359, -5.2584143666846e-6 - sin\13.6983894161836*\/ 5 / + cos\13.6983894161836*\/ 5 /)

                                                 /                  ___\      /                  ___\ 
(-27.748018187612004, -8.32694255623788e-12 - sin\27.748018187612*\/ 5 / + cos\27.748018187612*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-55.84727711177378, -1.04303989540769e-23 - sin\55.8472771117738*\/ 5 / + cos\55.8472771117738*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-41.79764764969232, -9.88536218608364e-18 - sin\41.7976476496923*\/ 5 / + cos\41.7976476496923*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-88.16142487456112, -1.51976329919925e-37 - sin\88.1614248745611*\/ 5 / + cos\88.1614248745611*\/ 5 /)

                                                 /                   ___\      /                   ___\ 
(-23.533129348921527, -4.79025444847835e-10 - sin\23.5331293489215*\/ 5 / + cos\23.5331293489215*\/ 5 /)

                                           /                   ___\      /                   ___\ 
(1.3377182562821472, 1.27568246847871 + cos\1.33771825628215*\/ 5 / + sin\1.33771825628215*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-93.7812766593937, -5.85928783607614e-40 - sin\93.7812766593937*\/ 5 / + cos\93.7812766593937*\/ 5 /)

                                               /                   ___\      /                   ___\ 
(-37.58275881106788, -6.0216782827729e-16 - sin\37.5827588110679*\/ 5 / + cos\37.5827588110679*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-36.17779586485974, -2.36312610918566e-15 - sin\36.1777958648597*\/ 5 / + cos\36.1777958648597*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-89.56638782076926, -3.78834127156158e-38 - sin\89.5663878207693*\/ 5 / + cos\89.5663878207693*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-97.99616549801813, -9.04416855119748e-42 - sin\97.9961654980181*\/ 5 / + cos\97.9961654980181*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-40.39268470348418, -3.89431088525322e-17 - sin\40.3926847034842*\/ 5 / + cos\40.3926847034842*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-92.37631371318555, -2.35223316395036e-39 - sin\92.3763137131856*\/ 5 / + cos\92.3763137131856*\/ 5 /)

                                                 /                   ___\      /                   ___\ 
(-34.772832918651595, -9.26003639807663e-15 - sin\34.7728329186516*\/ 5 / + cos\34.7728329186516*\/ 5 /)

                                                /                   ___\      /                   ___\ 
(-85.35149898214483, -2.44398010539843e-36 - sin\85.3514989821448*\/ 5 / + cos\85.3514989821448*\/ 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:
x1=−71.3018695200634x_{1} = -71.3018695200634
x2=−74.1117954124797x_{2} = -74.1117954124797
x3=−17.9132775506499x_{3} = -17.9132775506499
x4=−65.6820177352308x_{4} = -65.6820177352308
x5=−3.86047764079769x_{5} = -3.86047764079769
x6=−60.0621659503982x_{6} = -60.0621659503982
x7=−237.087497172625x_{7} = -237.087497172625
x8=−79.7316471973122x_{8} = -79.7316471973122
x9=−12.2934232680916x_{9} = -12.2934232680916
x10=−54.4423141655656x_{10} = -54.4423141655656
x11=−51.6323882731493x_{11} = -51.6323882731493
x12=−46.0125364883168x_{12} = -46.0125364883168
x13=−99.4011284442263x_{13} = -99.4011284442263
x14=−9.48346825780183x_{14} = -9.48346825780183
x15=−31.9629070262353x_{15} = -31.9629070262353
x16=−26.3430552413983x_{16} = -26.3430552413983
x17=−88.1614248745611x_{17} = -88.1614248745611
x18=−23.5331293489215x_{18} = -23.5331293489215
x19=1.33771825628215x_{19} = 1.33771825628215
x20=−93.7812766593937x_{20} = -93.7812766593937
x21=−37.5827588110679x_{21} = -37.5827588110679
x22=−40.3926847034842x_{22} = -40.3926847034842
x23=−34.7728329186516x_{23} = -34.7728329186516
x24=−85.3514989821448x_{24} = -85.3514989821448
Puntos máximos de la función:
x24=−64.2770547890226x_{24} = -64.2770547890226
x24=−75.5167583586878x_{24} = -75.5167583586878
x24=−78.3266842511041x_{24} = -78.3266842511041
x24=−19.3182405139455x_{24} = -19.3182405139455
x24=−83.9465360359367x_{24} = -83.9465360359367
x24=−33.3678699724435x_{24} = -33.3678699724435
x24=−8.07864528768309x_{24} = -8.07864528768309
x24=−69.8969065738552x_{24} = -69.8969065738552
x24=−30.5579440800272x_{24} = -30.5579440800272
x24=−61.4671288966064x_{24} = -61.4671288966064
x24=−47.4174994345249x_{24} = -47.4174994345249
x24=0.431345129440401x_{24} = 0.431345129440401
x24=−22.1281664030266x_{24} = -22.1281664030266
x24=−50.2274253269412x_{24} = -50.2274253269412
x24=−13.6983894161836x_{24} = -13.6983894161836
x24=−27.748018187612x_{24} = -27.748018187612
x24=−55.8472771117738x_{24} = -55.8472771117738
x24=−41.7976476496923x_{24} = -41.7976476496923
x24=−36.1777958648597x_{24} = -36.1777958648597
x24=−89.5663878207693x_{24} = -89.5663878207693
x24=−97.9961654980181x_{24} = -97.9961654980181
x24=−92.3763137131856x_{24} = -92.3763137131856
Decrece en los intervalos
[1.33771825628215,∞)\left[1.33771825628215, \infty\right)
Crece en los intervalos
(−∞,−237.087497172625]\left(-\infty, -237.087497172625\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
xex3+5ex9−5sin⁡(5x)−5cos⁡(5x)=0\frac{x e^{x}}{3} + \frac{5 e^{x}}{9} - 5 \sin{\left(\sqrt{5} x \right)} - 5 \cos{\left(\sqrt{5} x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−0.331010763507259x_{1} = -0.331010763507259
x2=−4.56549371377194x_{2} = -4.56549371377194
x3=−87.458943401457x_{3} = -87.458943401457
x4=−62.1696103697104x_{4} = -62.1696103697104
x5=−34.0703514455475x_{5} = -34.0703514455475
x6=−76.2192398317919x_{6} = -76.2192398317919
x7=−21.4256849294679x_{7} = -21.4256849294679
x8=−39.6902032303801x_{8} = -39.6902032303801
x9=−24.2356108220763x_{9} = -24.2356108220763
x10=−72.0043509931674x_{10} = -72.0043509931674
x11=−20.0207219842473x_{11} = -20.0207219842473
x12=−38.285240284172x_{12} = -38.285240284172
x13=−49.5249438538371x_{13} = -49.5249438538371
x14=−5.97132401247418x_{14} = -5.97132401247418
x15=−66.3844992083349x_{15} = -66.3844992083349
x16=−63.5745733159186x_{16} = -63.5745733159186
x17=−86.0539804552489x_{17} = -86.0539804552489
x18=−15.8058331041086x_{18} = -15.8058331041086
x19=0.915541600167856x_{19} = 0.915541600167856
x20=−25.6405737683023x_{20} = -25.6405737683023
x21=−7.37598010684999x_{21} = -7.37598010684999
x22=−59.3596844772941x_{22} = -59.3596844772941
x23=−14.4008703481407x_{23} = -14.4008703481407
x24=−73.4093139393756x_{24} = -73.4093139393756
x25=−35.4753143917557x_{25} = -35.4753143917557
x26=−32.6653884993394x_{26} = -32.6653884993394
x27=−100.10360991733x_{27} = -100.10360991733
x28=−29.855462606923x_{28} = -29.855462606923
x29=−43.9050920690045x_{29} = -43.9050920690045
x30=−95.8887210787059x_{30} = -95.8887210787059
x31=−48.119980907629x_{31} = -48.119980907629
x32=−80.4341286704163x_{32} = -80.4341286704163
x33=−57.954721531086x_{33} = -57.954721531086
x34=−28.4504996607152x_{34} = -28.4504996607152
x35=−1.7558787719912x_{35} = -1.7558787719912
x36=−91.6738322400815x_{36} = -91.6738322400815
x37=−107.128424648371x_{37} = -107.128424648371
x38=−67.789462154543x_{38} = -67.789462154543
x39=−45.3100550152127x_{39} = -45.3100550152127
x40=−11.5909462414078x_{40} = -11.5909462414078
x41=−77.624202778x_{41} = -77.624202778
x42=−81.8390916166245x_{42} = -81.8390916166245
x43=−10.1859745898246x_{43} = -10.1859745898246
x44=−53.7398326924616x_{44} = -53.7398326924616
x45=−90.2688692938733x_{45} = -90.2688692938733
x46=−52.3348697462534x_{46} = -52.3348697462534

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[0.915541600167856,∞)\left[0.915541600167856, \infty\right)
Convexa en los intervalos
(−∞,−100.10360991733]\left(-\infty, -100.10360991733\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((xex3+(−1)ex9)+cos⁡(5x))+sin⁡(5x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\left(\left(\frac{x e^{x}}{3} + \frac{\left(-1\right) e^{x}}{9}\right) + \cos{\left(\sqrt{5} x \right)}\right) + \sin{\left(\sqrt{5} x \right)}\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(((xex3+(−1)ex9)+cos⁡(5x))+sin⁡(5x))=∞\lim_{x \to \infty}\left(\left(\left(\frac{x e^{x}}{3} + \frac{\left(-1\right) e^{x}}{9}\right) + \cos{\left(\sqrt{5} x \right)}\right) + \sin{\left(\sqrt{5} x \right)}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-exp(x))/9 + (x*exp(x))/3 + cos(x*sqrt(5)) + sin(x*sqrt(5)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((xex3+(−1)ex9)+cos⁡(5x))+sin⁡(5x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\left(\frac{x e^{x}}{3} + \frac{\left(-1\right) e^{x}}{9}\right) + \cos{\left(\sqrt{5} x \right)}\right) + \sin{\left(\sqrt{5} x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(((xex3+(−1)ex9)+cos⁡(5x))+sin⁡(5x)x)=∞\lim_{x \to \infty}\left(\frac{\left(\left(\frac{x e^{x}}{3} + \frac{\left(-1\right) e^{x}}{9}\right) + \cos{\left(\sqrt{5} x \right)}\right) + \sin{\left(\sqrt{5} x \right)}}{x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
((xex3+(−1)ex9)+cos⁡(5x))+sin⁡(5x)=−xe−x3−sin⁡(5x)+cos⁡(5x)−e−x9\left(\left(\frac{x e^{x}}{3} + \frac{\left(-1\right) e^{x}}{9}\right) + \cos{\left(\sqrt{5} x \right)}\right) + \sin{\left(\sqrt{5} x \right)} = - \frac{x e^{- x}}{3} - \sin{\left(\sqrt{5} x \right)} + \cos{\left(\sqrt{5} x \right)} - \frac{e^{- x}}{9}
- No
((xex3+(−1)ex9)+cos⁡(5x))+sin⁡(5x)=xe−x3+sin⁡(5x)−cos⁡(5x)+e−x9\left(\left(\frac{x e^{x}}{3} + \frac{\left(-1\right) e^{x}}{9}\right) + \cos{\left(\sqrt{5} x \right)}\right) + \sin{\left(\sqrt{5} x \right)} = \frac{x e^{- x}}{3} + \sin{\left(\sqrt{5} x \right)} - \cos{\left(\sqrt{5} x \right)} + \frac{e^{- x}}{9}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор