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Trazado el gráfico de la función/ -((2/5)*(2)^x+6*(-(2/15)*cos((x*π)/2)+(1/10)*sin((x*π)/2)))

Gráfico de la función y = -((2/5)*(2)^x+6*(-(2/15)*cos((x*π)/2)+(1/10)*sin((x*π)/2)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                  /       /x*pi\      /x*pi\\
            x     |  2*cos|----|   sin|----||
         2*2      |       \ 2  /      \ 2  /|
f(x) = - ---- - 6*|- ----------- + ---------|
          5       \       15           10   /
f(x)=−2⋅2x5−6(sin⁡(πx2)10−2cos⁡(πx2)15)f{\left(x \right)} = - \frac{2 \cdot 2^{x}}{5} - 6 \left(\frac{\sin{\left(\frac{\pi x}{2} \right)}}{10} - \frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{15}\right) 
f = -2*2^x/5 - 6*(sin((pi*x)/2)/10 - 2*cos((pi*x)/2)/15)
Gráfico de la función
02468-8-6-4-2-1010-500500
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:
−2⋅2x5−6(sin⁡(πx2)10−2cos⁡(πx2)15)=0- \frac{2 \cdot 2^{x}}{5} - 6 \left(\frac{\sin{\left(\frac{\pi x}{2} \right)}}{10} - \frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{15}\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−81.4096655293983x_{1} = -81.4096655293983
x2=−93.4096655293983x_{2} = -93.4096655293983
x3=−5.40364983325559x_{3} = -5.40364983325559
x4=−47.4096655293983x_{4} = -47.4096655293983
x5=−85.4096655293983x_{5} = -85.4096655293983
x6=−57.4096655293983x_{6} = -57.4096655293983
x7=−79.4096655293983x_{7} = -79.4096655293983
x8=0.269936198321321x_{8} = 0.269936198321321
x9=−31.4096655294875x_{9} = -31.4096655294875
x10=−69.4096655293983x_{10} = -69.4096655293983
x11=−67.4096655293983x_{11} = -67.4096655293983
x12=−91.4096655293983x_{12} = -91.4096655293983
x13=−49.4096655293983x_{13} = -49.4096655293983
x14=−11.4097591260635x_{14} = -11.4097591260635
x15=−77.4096655293983x_{15} = -77.4096655293983
x16=−39.4096655293986x_{16} = -39.4096655293986
x17=−15.4096713795456x_{17} = -15.4096713795456
x18=−25.4096655236852x_{18} = -25.4096655236852
x19=−89.4096655293983x_{19} = -89.4096655293983
x20=−29.4096655290412x_{20} = -29.4096655290412
x21=−45.4096655293983x_{21} = -45.4096655293983
x22=−7.41116162230209x_{22} = -7.41116162230209
x23=−17.409664066854x_{23} = -17.409664066854
x24=−5.40364983325559x_{24} = -5.40364983325559
x25=−63.4096655293983x_{25} = -63.4096655293983
x26=−35.4096655294038x_{26} = -35.4096655294038
x27=−53.4096655293983x_{27} = -53.4096655293983
x28=−55.4096655293983x_{28} = -55.4096655293983
x29=−65.4096655293983x_{29} = -65.4096655293983
x30=−19.4096658950339x_{30} = -19.4096658950339
x31=−99.4096655293983x_{31} = -99.4096655293983
x32=−21.4096654379893x_{32} = -21.4096654379893
x33=−51.4096655293983x_{33} = -51.4096655293983
x34=−41.4096655293982x_{34} = -41.4096655293982
x35=−97.4096655293983x_{35} = -97.4096655293983
x36=−1.30623817628826x_{36} = -1.30623817628826
x37=−37.4096655293969x_{37} = -37.4096655293969
x38=−13.4096421283344x_{38} = -13.4096421283344
x39=−61.4096655293983x_{39} = -61.4096655293983
x40=−23.4096655522505x_{40} = -23.4096655522505
x41=−59.4096655293983x_{41} = -59.4096655293983
x42=−71.4096655293983x_{42} = -71.4096655293983
x43=−9.40929102122168x_{43} = -9.40929102122168
x44=−75.4096655293983x_{44} = -75.4096655293983
x45=−73.4096655293983x_{45} = -73.4096655293983
x46=−87.4096655293983x_{46} = -87.4096655293983
x47=−3.4332447669414x_{47} = -3.4332447669414
x48=−83.4096655293983x_{48} = -83.4096655293983
x49=−33.4096655293759x_{49} = -33.4096655293759
x50=−43.4096655293983x_{50} = -43.4096655293983
x51=−27.4096655308265x_{51} = -27.4096655308265
x52=−95.4096655293983x_{52} = -95.4096655293983
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 -2*2^x/5 - 6*(-2*cos((x*pi)/2)/15 + sin((x*pi)/2)/10).
−2⋅205−6(−2cos⁡(0π2)15+sin⁡(0π2)10)- \frac{2 \cdot 2^{0}}{5} - 6 \left(- \frac{2 \cos{\left(\frac{0 \pi}{2} \right)}}{15} + \frac{\sin{\left(\frac{0 \pi}{2} \right)}}{10}\right)
Resultado:
f(0)=25f{\left(0 \right)} = \frac{2}{5}
Punto:
(0, 2/5)
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
−2⋅2xlog⁡(2)5−2πsin⁡(πx2)5−3πcos⁡(πx2)10=0- \frac{2 \cdot 2^{x} \log{\left(2 \right)}}{5} - \frac{2 \pi \sin{\left(\frac{\pi x}{2} \right)}}{5} - \frac{3 \pi \cos{\left(\frac{\pi x}{2} \right)}}{10} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−12.4096861811988x_{1} = -12.4096861811988
x2=−82.4096655293983x_{2} = -82.4096655293983
x3=−100.409665529398x_{3} = -100.409665529398
x4=−4.41493325707022x_{4} = -4.41493325707022
x5=−8.40999588729489x_{5} = -8.40999588729489
x6=−68.4096655293983x_{6} = -68.4096655293983
x7=−92.4096655293983x_{7} = -92.4096655293983
x8=−40.4096655293983x_{8} = -40.4096655293983
x9=−30.4096655293195x_{9} = -30.4096655293195
x10=−88.4096655293983x_{10} = -88.4096655293983
x11=−16.4096668201531x_{11} = -16.4096668201531
x12=2.18125121798966x_{12} = 2.18125121798966
x13=−74.4096655293983x_{13} = -74.4096655293983
x14=−66.4096655293983x_{14} = -66.4096655293983
x15=−20.4096656100705x_{15} = -20.4096656100705
x16=−78.4096655293983x_{16} = -78.4096655293983
x17=−96.4096655293983x_{17} = -96.4096655293983
x18=−26.4096655281378x_{18} = -26.4096655281378
x19=−48.4096655293983x_{19} = -48.4096655293983
x20=−2.38819665927311x_{20} = -2.38819665927311
x21=−60.4096655293983x_{21} = -60.4096655293983
x22=−86.4096655293983x_{22} = -86.4096655293983
x23=−38.409665529398x_{23} = -38.409665529398
x24=−70.4096655293983x_{24} = -70.4096655293983
x25=−28.4096655297134x_{25} = -28.4096655297134
x26=−72.4096655293983x_{26} = -72.4096655293983
x27=−0.489893043321873x_{27} = -0.489893043321873
x28=−80.4096655293983x_{28} = -80.4096655293983
x29=−56.4096655293983x_{29} = -56.4096655293983
x30=−94.4096655293983x_{30} = -94.4096655293983
x31=−32.409665529418x_{31} = -32.409665529418
x32=−52.4096655293983x_{32} = -52.4096655293983
x33=−34.4096655293933x_{33} = -34.4096655293933
x34=−50.4096655293983x_{34} = -50.4096655293983
x35=−90.4096655293983x_{35} = -90.4096655293983
x36=−58.4096655293983x_{36} = -58.4096655293983
x37=−46.4096655293983x_{37} = -46.4096655293983
x38=−42.4096655293982x_{38} = -42.4096655293982
x39=−36.4096655293995x_{39} = -36.4096655293995
x40=−64.4096655293983x_{40} = -64.4096655293983
x41=−6.40834258171133x_{41} = -6.40834258171133
x42=−54.4096655293983x_{42} = -54.4096655293983
x43=−10.409582916283x_{43} = -10.409582916283
x44=−44.4096655293983x_{44} = -44.4096655293983
x45=−18.4096652067092x_{45} = -18.4096652067092
x46=−24.4096655344403x_{46} = -24.4096655344403
x47=−98.4096655293983x_{47} = -98.4096655293983
x48=2.48068459959753x_{48} = 2.48068459959753
x49=−76.4096655293983x_{49} = -76.4096655293983
x50=−84.4096655293983x_{50} = -84.4096655293983
x51=−14.4096603663558x_{51} = -14.4096603663558
x52=−62.4096655293983x_{52} = -62.4096655293983
x53=−22.4096655092302x_{53} = -22.4096655092302
Signos de extremos en los puntos:
                                            3*sin(0.204843090599384*pi)   4*cos(0.204843090599384*pi) 
(-12.409686181198769, -7.3514365636572e-5 + --------------------------- + ---------------------------)
                                                         5                             5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-82.40966552939827, -6.22699811627173e-26 + -------------------------- + --------------------------)
                                                         5                            5              

                                              3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-100.40966552939827, -2.37541126871938e-31 + --------------------------- + ---------------------------)
                                                           5                             5              

                                           3*sin(0.20746662853511*pi)   4*cos(0.20746662853511*pi) 
(-4.414933257070221, -0.0187513548338151 + -------------------------- + --------------------------)
                                                       5                            5              

                                            3*sin(0.204997943647447*pi)   4*cos(0.204997943647447*pi) 
(-8.409995887294894, -0.00117597737378052 + --------------------------- + ---------------------------)
                                                         5                             5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-68.40966552939827, -1.02023137136996e-21 + --------------------------- + ---------------------------)
                                                          5                             5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-92.40966552939827, -6.08105284792161e-29 + --------------------------- + ---------------------------)
                                                          5                             5              

                                              3*sin(0.204832764699173*pi)   4*cos(0.204832764699173*pi) 
(-40.409665529398346, -2.73866273399186e-13 + --------------------------- + ---------------------------)
                                                           5                             5              

                                              3*sin(1.20483276465974*pi)   4*cos(1.20483276465974*pi) 
(-30.409665529319486, -2.80439063976096e-10 + -------------------------- + --------------------------)
                                                          5                            5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-88.40966552939827, -9.72968455667458e-28 + --------------------------- + ---------------------------)
                                                          5                             5              

                                            3*sin(0.20483341007656*pi)   4*cos(0.20483341007656*pi) 
(-16.40966682015312, -4.59470951312272e-6 + -------------------------- + --------------------------)
                                                        5                            5              

                                        3*sin(1.09062560899483*pi)   4*cos(1.09062560899483*pi) 
(2.181251217989658, -1.81418693920325 - -------------------------- + --------------------------)
                                                    5                            5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-74.40966552939827, -1.59411151776556e-23 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-66.40966552939827, -4.08092548547984e-21 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(0.204832805035256*pi)   4*cos(0.204832805035256*pi) 
(-20.409665610070512, -2.87169585437966e-7 + --------------------------- + ---------------------------)
                                                          5                             5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-78.40966552939827, -9.96319698603477e-25 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-96.40966552939827, -3.80065802995101e-30 + --------------------------- + ---------------------------)
                                                          5                             5              

                                            3*sin(1.20483276406888*pi)   4*cos(1.20483276406888*pi) 
(-26.40966552813776, -4.48702502729289e-9 + -------------------------- + --------------------------)
                                                        5                            5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-48.40966552939827, -1.06979013046563e-15 + --------------------------- + ---------------------------)
                                                          5                             5              

                                           3*sin(1.19409832963656*pi)   4*cos(1.19409832963656*pi) 
(-2.388196659273115, -0.0764084098060627 + -------------------------- + --------------------------)
                                                       5                            5              

                                            3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-60.40966552939827, -2.6117923107071e-19 + --------------------------- + ---------------------------)
                                                         5                             5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-86.40966552939827, -3.89187382266983e-27 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(1.20483276469898*pi)   4*cos(1.20483276469898*pi) 
(-38.40966552939796, -1.09546509359704e-12 + -------------------------- + --------------------------)
                                                         5                            5              

                                            3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-70.40966552939827, -2.5505784284249e-22 + -------------------------- + --------------------------)
                                                        5                            5              

                                            3*sin(0.204832764856697*pi)   4*cos(0.204832764856697*pi) 
(-28.409665529713394, -1.1217562555981e-9 + --------------------------- + ---------------------------)
                                                         5                             5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-72.40966552939827, -6.37644607106226e-23 + --------------------------- + ---------------------------)
                                                          5                             5              

                                            3*sin(0.244946521660937*pi)   4*cos(0.244946521660937*pi) 
(-0.48989304332187333, -0.284831154784219 + --------------------------- + ---------------------------)
                                                         5                             5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-80.40966552939827, -2.49079924650869e-25 + --------------------------- + ---------------------------)
                                                          5                             5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-56.40966552939827, -4.17886769713136e-18 + --------------------------- + ---------------------------)
                                                          5                             5              

                                            3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-94.40966552939827, -1.5202632119804e-29 + -------------------------- + --------------------------)
                                                        5                            5              

                                              3*sin(0.204832764708982*pi)   4*cos(0.204832764708982*pi) 
(-32.409665529417964, -7.01097659892382e-11 + --------------------------- + ---------------------------)
                                                           5                             5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-52.40966552939827, -6.68618831541018e-17 + --------------------------- + ---------------------------)
                                                          5                             5              

                                              3*sin(1.20483276469667*pi)   4*cos(1.20483276469667*pi) 
(-34.409665529393344, -1.75274414976087e-11 + -------------------------- + --------------------------)
                                                          5                            5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-50.40966552939827, -2.67447532616407e-16 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-90.40966552939827, -2.43242113916865e-28 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-58.40966552939827, -1.04471692428284e-18 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-46.40966552939827, -4.27916052186251e-15 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(1.20483276469912*pi)   4*cos(1.20483276469912*pi) 
(-42.40966552939825, -6.84665683498012e-14 + -------------------------- + --------------------------)
                                                         5                            5              

                                            3*sin(0.204832764699749*pi)   4*cos(0.204832764699749*pi) 
(-36.4096655293995, -4.38186037438348e-12 + --------------------------- + ---------------------------)
                                                         5                             5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-64.40966552939827, -1.63237019419194e-20 + --------------------------- + ---------------------------)
                                                          5                             5              

                                           3*sin(1.20417129085566*pi)   4*cos(1.20417129085566*pi) 
(-6.40834258171133, -0.00470930319058378 + -------------------------- + --------------------------)
                                                       5                            5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-54.40966552939827, -1.67154707885254e-17 + -------------------------- + --------------------------)
                                                         5                            5              

                                              3*sin(1.2047914581415*pi)   4*cos(1.2047914581415*pi) 
(-10.409582916282991, -0.000294078511281487 + ------------------------- + -------------------------)
                                                          5                           5             

                                            3*sin(0.204832764699137*pi)   4*cos(0.204832764699137*pi) 
(-44.409665529398275, -1.711664208745e-14 + --------------------------- + ---------------------------)
                                                         5                             5              

                                             3*sin(1.2048326033546*pi)   4*cos(1.2048326033546*pi) 
(-18.409665206709192, -1.14867866290946e-6 + ------------------------- + -------------------------)
                                                         5                           5             

                                            3*sin(0.204832767220141*pi)   4*cos(0.204832767220141*pi) 
(-24.409665534440283, -1.7948100030764e-8 + --------------------------- + ---------------------------)
                                                         5                             5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-98.40966552939827, -9.50164507487752e-31 + -------------------------- + --------------------------)
                                                         5                            5              

                                        3*sin(1.24034229979877*pi)   4*cos(1.24034229979877*pi) 
(2.480684599597533, -2.23264906995459 - -------------------------- + --------------------------)
                                                    5                            5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-76.40966552939827, -3.98527879441391e-24 + --------------------------- + ---------------------------)
                                                          5                             5              

                                             3*sin(0.204832764699134*pi)   4*cos(0.204832764699134*pi) 
(-84.40966552939827, -1.55674952906793e-26 + --------------------------- + ---------------------------)
                                                          5                             5              

                                             3*sin(1.20483018317788*pi)   4*cos(1.20483018317788*pi) 
(-14.409660366355759, -1.83789202691468e-5 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(1.20483276469913*pi)   4*cos(1.20483276469913*pi) 
(-62.40966552939827, -6.52948077676775e-20 + -------------------------- + --------------------------)
                                                         5                            5              

                                             3*sin(1.2048327546151*pi)   4*cos(1.2048327546151*pi) 
(-22.409665509230205, -7.17924013775774e-8 + ------------------------- + -------------------------)
                                                         5                           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=−82.4096655293983x_{1} = -82.4096655293983
x2=−30.4096655293195x_{2} = -30.4096655293195
x3=2.18125121798966x_{3} = 2.18125121798966
x4=−74.4096655293983x_{4} = -74.4096655293983
x5=−66.4096655293983x_{5} = -66.4096655293983
x6=−78.4096655293983x_{6} = -78.4096655293983
x7=−26.4096655281378x_{7} = -26.4096655281378
x8=−2.38819665927311x_{8} = -2.38819665927311
x9=−86.4096655293983x_{9} = -86.4096655293983
x10=−38.409665529398x_{10} = -38.409665529398
x11=−70.4096655293983x_{11} = -70.4096655293983
x12=−94.4096655293983x_{12} = -94.4096655293983
x13=−34.4096655293933x_{13} = -34.4096655293933
x14=−50.4096655293983x_{14} = -50.4096655293983
x15=−90.4096655293983x_{15} = -90.4096655293983
x16=−58.4096655293983x_{16} = -58.4096655293983
x17=−46.4096655293983x_{17} = -46.4096655293983
x18=−42.4096655293982x_{18} = -42.4096655293982
x19=−6.40834258171133x_{19} = -6.40834258171133
x20=−54.4096655293983x_{20} = -54.4096655293983
x21=−10.409582916283x_{21} = -10.409582916283
x22=−18.4096652067092x_{22} = -18.4096652067092
x23=−98.4096655293983x_{23} = -98.4096655293983
x24=−14.4096603663558x_{24} = -14.4096603663558
x25=−62.4096655293983x_{25} = -62.4096655293983
x26=−22.4096655092302x_{26} = -22.4096655092302
Puntos máximos de la función:
x26=−12.4096861811988x_{26} = -12.4096861811988
x26=−100.409665529398x_{26} = -100.409665529398
x26=−4.41493325707022x_{26} = -4.41493325707022
x26=−8.40999588729489x_{26} = -8.40999588729489
x26=−68.4096655293983x_{26} = -68.4096655293983
x26=−92.4096655293983x_{26} = -92.4096655293983
x26=−40.4096655293983x_{26} = -40.4096655293983
x26=−88.4096655293983x_{26} = -88.4096655293983
x26=−16.4096668201531x_{26} = -16.4096668201531
x26=−20.4096656100705x_{26} = -20.4096656100705
x26=−96.4096655293983x_{26} = -96.4096655293983
x26=−48.4096655293983x_{26} = -48.4096655293983
x26=−60.4096655293983x_{26} = -60.4096655293983
x26=−28.4096655297134x_{26} = -28.4096655297134
x26=−72.4096655293983x_{26} = -72.4096655293983
x26=−0.489893043321873x_{26} = -0.489893043321873
x26=−80.4096655293983x_{26} = -80.4096655293983
x26=−56.4096655293983x_{26} = -56.4096655293983
x26=−32.409665529418x_{26} = -32.409665529418
x26=−52.4096655293983x_{26} = -52.4096655293983
x26=−36.4096655293995x_{26} = -36.4096655293995
x26=−64.4096655293983x_{26} = -64.4096655293983
x26=−44.4096655293983x_{26} = -44.4096655293983
x26=−24.4096655344403x_{26} = -24.4096655344403
x26=2.48068459959753x_{26} = 2.48068459959753
x26=−76.4096655293983x_{26} = -76.4096655293983
x26=−84.4096655293983x_{26} = -84.4096655293983
Decrece en los intervalos
[2.18125121798966,∞)\left[2.18125121798966, \infty\right)
Crece en los intervalos
(−∞,−98.4096655293983]\left(-\infty, -98.4096655293983\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
−8⋅2xlog⁡(2)2+3π2sin⁡(πx2)−4π2cos⁡(πx2)20=0\frac{- 8 \cdot 2^{x} \log{\left(2 \right)}^{2} + 3 \pi^{2} \sin{\left(\frac{\pi x}{2} \right)} - 4 \pi^{2} \cos{\left(\frac{\pi x}{2} \right)}}{20} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−99.4096655293983x_{1} = -99.4096655293983
x2=−95.4096655293983x_{2} = -95.4096655293983
x3=2.33350159222128x_{3} = 2.33350159222128
x4=−77.4096655293983x_{4} = -77.4096655293983
x5=−5.41083107446377x_{5} = -5.41083107446377
x6=−7.40937384881488x_{6} = -7.40937384881488
x7=−59.4096655293983x_{7} = -59.4096655293983
x8=−85.4096655293983x_{8} = -85.4096655293983
x9=−91.4096655293983x_{9} = -91.4096655293983
x10=−79.4096655293983x_{10} = -79.4096655293983
x11=−35.4096655293972x_{11} = -35.4096655293972
x12=−93.4096655293983x_{12} = -93.4096655293983
x13=−65.4096655293983x_{13} = -65.4096655293983
x14=−63.4096655293983x_{14} = -63.4096655293983
x15=−55.4096655293983x_{15} = -55.4096655293983
x16=−17.4096658141849x_{16} = -17.4096658141849
x17=0.669398184051839x_{17} = 0.669398184051839
x18=−27.4096655291202x_{18} = -27.4096655291202
x19=−41.4096655293983x_{19} = -41.4096655293983
x20=−1.4280949854068x_{20} = -1.4280949854068
x21=−83.4096655293983x_{21} = -83.4096655293983
x22=−37.4096655293985x_{22} = -37.4096655293985
x23=−71.4096655293983x_{23} = -71.4096655293983
x24=−39.4096655293982x_{24} = -39.4096655293982
x25=−31.4096655293809x_{25} = -31.4096655293809
x26=−97.4096655293983x_{26} = -97.4096655293983
x27=−51.4096655293983x_{27} = -51.4096655293983
x28=−67.4096655293983x_{28} = -67.4096655293983
x29=−3.40498437717754x_{29} = -3.40498437717754
x30=−53.4096655293983x_{30} = -53.4096655293983
x31=−49.4096655293983x_{31} = -49.4096655293983
x32=−23.4096655249485x_{32} = -23.4096655249485
x33=−11.4096473028175x_{33} = -11.4096473028175
x34=−75.4096655293983x_{34} = -75.4096655293983
x35=−21.4096655471974x_{35} = -21.4096655471974
x36=−25.4096655305107x_{36} = -25.4096655305107
x37=−15.4096643902505x_{37} = -15.4096643902505
x38=−47.4096655293983x_{38} = -47.4096655293983
x39=−45.4096655293983x_{39} = -45.4096655293983
x40=−81.4096655293983x_{40} = -81.4096655293983
x41=−43.4096655293983x_{41} = -43.4096655293983
x42=−13.4096700859715x_{42} = -13.4096700859715
x43=−57.4096655293983x_{43} = -57.4096655293983
x44=−73.4096655293983x_{44} = -73.4096655293983
x45=−61.4096655293983x_{45} = -61.4096655293983
x46=−89.4096655293983x_{46} = -89.4096655293983
x47=−33.4096655294026x_{47} = -33.4096655294026
x48=−69.4096655293983x_{48} = -69.4096655293983
x49=−19.4096654582016x_{49} = -19.4096654582016
x50=−87.4096655293983x_{50} = -87.4096655293983
x51=−9.40973843111646x_{51} = -9.40973843111646
x52=−29.4096655294678x_{52} = -29.4096655294678

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
[0.669398184051839,∞)\left[0.669398184051839, \infty\right)
Convexa en los intervalos
(−∞,−99.4096655293983]\left(-\infty, -99.4096655293983\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−2⋅2x5−6(sin⁡(πx2)10−2cos⁡(πx2)15))=⟨−75,75⟩\lim_{x \to -\infty}\left(- \frac{2 \cdot 2^{x}}{5} - 6 \left(\frac{\sin{\left(\frac{\pi x}{2} \right)}}{10} - \frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{15}\right)\right) = \left\langle - \frac{7}{5}, \frac{7}{5}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−75,75⟩y = \left\langle - \frac{7}{5}, \frac{7}{5}\right\rangle
lim⁡x→∞(−2⋅2x5−6(sin⁡(πx2)10−2cos⁡(πx2)15))=−∞\lim_{x \to \infty}\left(- \frac{2 \cdot 2^{x}}{5} - 6 \left(\frac{\sin{\left(\frac{\pi x}{2} \right)}}{10} - \frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{15}\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 -2*2^x/5 - 6*(-2*cos((x*pi)/2)/15 + sin((x*pi)/2)/10), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−2⋅2x5−6(sin⁡(πx2)10−2cos⁡(πx2)15)x)=0\lim_{x \to -\infty}\left(\frac{- \frac{2 \cdot 2^{x}}{5} - 6 \left(\frac{\sin{\left(\frac{\pi x}{2} \right)}}{10} - \frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{15}\right)}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−2⋅2x5−6(sin⁡(πx2)10−2cos⁡(πx2)15)x)=−∞\lim_{x \to \infty}\left(\frac{- \frac{2 \cdot 2^{x}}{5} - 6 \left(\frac{\sin{\left(\frac{\pi x}{2} \right)}}{10} - \frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{15}\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:
−2⋅2x5−6(sin⁡(πx2)10−2cos⁡(πx2)15)=3sin⁡(πx2)5+4cos⁡(πx2)5−2⋅2−x5- \frac{2 \cdot 2^{x}}{5} - 6 \left(\frac{\sin{\left(\frac{\pi x}{2} \right)}}{10} - \frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{15}\right) = \frac{3 \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{4 \cos{\left(\frac{\pi x}{2} \right)}}{5} - \frac{2 \cdot 2^{- x}}{5}
- No
−2⋅2x5−6(sin⁡(πx2)10−2cos⁡(πx2)15)=−3sin⁡(πx2)5−4cos⁡(πx2)5+2⋅2−x5- \frac{2 \cdot 2^{x}}{5} - 6 \left(\frac{\sin{\left(\frac{\pi x}{2} \right)}}{10} - \frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{15}\right) = - \frac{3 \sin{\left(\frac{\pi x}{2} \right)}}{5} - \frac{4 \cos{\left(\frac{\pi x}{2} \right)}}{5} + \frac{2 \cdot 2^{- x}}{5}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор