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x*(-3*x/4+x*log(x)/2)*exp(-x)
Trazado el gráfico de la función/ x*(-3*x/4+x*log(x)/2)*exp(-x)

Gráfico de la función y = x*(-3*x/4+x*log(x)/2)*exp(-x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         /-3*x   x*log(x)\  -x
f(x) = x*|---- + --------|*e  
         \ 4        2    /
f(x)=x((−1)3x4+xlog⁡(x)2)e−xf{\left(x \right)} = x \left(\frac{\left(-1\right) 3 x}{4} + \frac{x \log{\left(x \right)}}{2}\right) e^{- x} 
f = (x*((-3*x)/4 + (x*log(x))/2))*exp(-x)
Gráfico de la función
02468-8-6-4-2-10100.5-0.5
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:
x((−1)3x4+xlog⁡(x)2)e−x=0x \left(\frac{\left(-1\right) 3 x}{4} + \frac{x \log{\left(x \right)}}{2}\right) e^{- x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=e32x_{1} = e^{\frac{3}{2}}
Solución numérica
x1=43.1554303825171x_{1} = 43.1554303825171
x2=99.7548364126332x_{2} = 99.7548364126332
x3=77.9661589714896x_{3} = 77.9661589714896
x4=50.656058037749x_{4} = 50.656058037749
x5=62.2524454631045x_{5} = 62.2524454631045
x6=35.3060631719167x_{6} = 35.3060631719167
x7=93.800350406415x_{7} = 93.800350406415
x8=56.4221156647367x_{8} = 56.4221156647367
x9=121.632858204347x_{9} = 121.632858204347
x10=101.741108654609x_{10} = 101.741108654609
x11=34.7171802642937x_{11} = 34.7171802642937
x12=89.8349023291442x_{12} = 89.8349023291442
x13=81.9169448994292x_{13} = 81.9169448994292
x14=58.3598729209293x_{14} = 58.3598729209293
x15=111.681226973085x_{15} = 111.681226973085
x16=64.2057597646063x_{16} = 64.2057597646063
x17=97.7692499352616x_{17} = 97.7692499352616
x18=119.641773175182x_{18} = 119.641773175182
x19=109.692162373785x_{19} = 109.692162373785
x20=36.2123527825047x_{20} = 36.2123527825047
x21=79.940759130625x_{21} = 79.940759130625
x22=83.8945721534703x_{22} = 83.8945721534703
x23=54.4913318425192x_{23} = 54.4913318425192
x24=74.0223953496608x_{24} = 74.0223953496608
x25=75.9933085008851x_{25} = 75.9933085008851
x26=41.3401550546804x_{26} = 41.3401550546804
x27=91.8171600140963x_{27} = 91.8171600140963
x28=87.8536571742414x_{28} = 87.8536571742414
x29=46.8688779662536x_{29} = 46.8688779662536
x30=103.728018905469x_{30} = 103.728018905469
x31=85.8735137678361x_{31} = 85.8735137678361
x32=52.5687821844656x_{32} = 52.5687821844656
x33=66.1629714263002x_{33} = 66.1629714263002
x34=68.1236095879084x_{34} = 68.1236095879084
x35=39.5651421324051x_{35} = 39.5651421324051
x36=95.7844018711505x_{36} = 95.7844018711505
x37=113.670747101399x_{37} = 113.670747101399
x38=48.7552006415197x_{38} = 48.7552006415197
x39=37.8466111913384x_{39} = 37.8466111913384
x40=72.0536351113193x_{40} = 72.0536351113193
x41=105.71552374607x_{41} = 105.71552374607
x42=4.48168907033806x_{42} = 4.48168907033806
x43=60.3035907653443x_{43} = 60.3035907653443
x44=115.660694920385x_{44} = 115.660694920385
x45=107.703583611838x_{45} = 107.703583611838
x46=45.0006557863195x_{46} = 45.0006557863195
x47=70.0872767904239x_{47} = 70.0872767904239
x48=117.651044814888x_{48} = 117.651044814888
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 (x*((-3*x)/4 + (x*log(x))/2))*exp(-x).
0(0log⁡(0)2+(−1)0⋅34)e−00 \left(\frac{0 \log{\left(0 \right)}}{2} + \frac{\left(-1\right) 0 \cdot 3}{4}\right) e^{- 0}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
−x((−1)3x4+xlog⁡(x)2)e−x+((−1)3x4+x(log⁡(x)2−14)+xlog⁡(x)2)e−x=0- x \left(\frac{\left(-1\right) 3 x}{4} + \frac{x \log{\left(x \right)}}{2}\right) e^{- x} + \left(\frac{\left(-1\right) 3 x}{4} + x \left(\frac{\log{\left(x \right)}}{2} - \frac{1}{4}\right) + \frac{x \log{\left(x \right)}}{2}\right) e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=41.454126947819x_{1} = 41.454126947819
x2=50.7060454108238x_{2} = 50.7060454108238
x3=36.4685968712191x_{3} = 36.4685968712191
x4=66.1844766504484x_{4} = 66.1844766504484
x5=60.331902638916x_{5} = 60.331902638916
x6=119.646419134701x_{6} = 119.646419134701
x7=105.721787493665x_{7} = 105.721787493665
x8=34.418598867512x_{8} = 34.418598867512
x9=107.709568738039x_{9} = 107.709568738039
x10=99.7620633877017x_{10} = 99.7620633877017
x11=89.8443102640912x_{11} = 89.8443102640912
x12=64.2292299517156x_{12} = 64.2292299517156
x13=70.1055291173656x_{13} = 70.1055291173656
x14=121.637325226112x_{14} = 121.637325226112
x15=83.9057987360247x_{15} = 83.9057987360247
x16=95.792399818575x_{16} = 95.792399818575
x17=93.8087809197351x_{17} = 93.8087809197351
x18=91.8260591469055x_{18} = 91.8260591469055
x19=43.2488900235804x_{19} = 43.2488900235804
x20=5.82199697552675x_{20} = 5.82199697552675
x21=77.9797881718651x_{21} = 77.9797881718651
x22=39.7079219594488x_{22} = 39.7079219594488
x23=76.0079125655966x_{23} = 76.0079125655966
x24=35.1122610868937x_{24} = 35.1122610868937
x25=68.1433879318773x_{25} = 68.1433879318773
x26=52.6127606020756x_{26} = 52.6127606020756
x27=74.0380830475647x_{27} = 74.0380830475647
x28=81.9288963522112x_{28} = 81.9288963522112
x29=97.7768477595718x_{29} = 97.7768477595718
x30=103.734581166979x_{30} = 103.734581166979
x31=58.3911943689988x_{31} = 58.3911943689988
x32=46.9354169943929x_{32} = 46.9354169943929
x33=101.747991263803x_{33} = 101.747991263803
x34=54.5303391811328x_{34} = 54.5303391811328
x35=85.8840795357641x_{35} = 85.8840795357641
x36=109.697887041197x_{36} = 109.697887041197
x37=111.686707798667x_{37} = 111.686707798667
x38=45.0788783769549x_{38} = 45.0788783769549
x39=113.675999317639x_{39} = 113.675999317639
x40=72.0705321145401x_{40} = 72.0705321145401
x41=62.2781650661523x_{41} = 62.2781650661523
x42=38.0323621111837x_{42} = 38.0323621111837
x43=48.8125551025324x_{43} = 48.8125551025324
x44=117.655880661938x_{44} = 117.655880661938
x45=87.8636188335214x_{45} = 87.8636188335214
x46=115.665732516682x_{46} = 115.665732516682
x47=56.4569601925841x_{47} = 56.4569601925841
x48=79.9535080952658x_{48} = 79.9535080952658
Signos de extremos en los puntos:
(41.45412694781901, 1.89695245512511e-15)

(50.70604541082377, 2.9691534493838e-19)

(36.4685968712191, 2.02386972946624e-13)

(66.18447665044842, 1.0643299376345e-25)

(60.33190263891597, 2.97304705387026e-23)

(119.64641913470109, 2.56722821688205e-48)

(105.7217874936648, 2.15133776678123e-42)

(34.41859886751195, 1.36169616224517e-12)

(107.7095687380389, 3.07721362019096e-43)

(99.7620633877017, 7.28684143206021e-40)

(89.84431026409119, 1.15852644915084e-35)

(64.2292299517156, 7.0035481869906e-25)

(70.10552911736558, 2.41748093378637e-27)

(121.63732522611208, 3.64196598037054e-49)

(83.9057987360247, 3.74585476503826e-33)

(95.79239981857498, 3.51198499960047e-38)

(93.80878091973507, 2.43150077266906e-37)

(91.82605914690548, 1.6801309013643e-36)

(43.248890023580365, 3.49632442223709e-16)

(5.821996975526752, 0.0131329549994981)

(77.97978817186507, 1.18186801482532e-30)

(39.707921959448775, 9.78494169243651e-15)

(76.0079125655966, 7.99439382827685e-30)

(35.112261086893746, 7.15134719533547e-13)

(68.14338793187733, 1.60822438285395e-26)

(52.612760602075575, 4.82145040918107e-20)

(74.03808304756468, 5.38787098409362e-29)

(81.92889635221115, 2.55770276912959e-32)

(97.77684775957184, 5.06321272904388e-39)

(103.73458116697917, 1.50191164175524e-41)

(58.391194368998846, 1.91489521966195e-22)

(46.93541699439291, 1.06910342763063e-17)

(101.74799126380286, 1.04696668196445e-40)

(54.53033918113279, 7.72227144248535e-21)

(85.88407953576414, 5.47112619476792e-34)

(109.69788704119651, 4.39563190016961e-44)

(111.68670779866709, 6.27086280333015e-45)

(45.07887837695488, 6.2047136243395e-17)

(113.67599931763876, 8.93514003215054e-46)

(72.07053211454014, 3.61682782540023e-28)

(62.27816506615228, 4.57930809142031e-24)

(38.03236211118371, 4.70032786143945e-14)

(48.81255510253243, 1.79904413614073e-18)

(117.65588066193791, 1.80779110111213e-47)

(87.86361883352137, 7.97085720366942e-35)

(115.66573251668186, 1.27164981770203e-46)

(56.456960192584084, 1.22229474946621e-21)

(79.95350809526585, 1.7413579768055e-31)


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:
La función no tiene puntos mínimos
Puntos máximos de la función:
x48=5.82199697552675x_{48} = 5.82199697552675
Decrece en los intervalos
(−∞,5.82199697552675]\left(-\infty, 5.82199697552675\right]
Crece en los intervalos
[5.82199697552675,∞)\left[5.82199697552675, \infty\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
(x2(2log⁡(x)−3)4−2x(log⁡(x)−1)+log⁡(x))e−x=0\left(\frac{x^{2} \left(2 \log{\left(x \right)} - 3\right)}{4} - 2 x \left(\log{\left(x \right)} - 1\right) + \log{\left(x \right)}\right) e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=85.8949773279601x_{1} = 85.8949773279601
x2=109.703742401224x_{2} = 109.703742401224
x3=0.300644050042844x_{3} = 0.300644050042844
x4=35.6998166830064x_{4} = 35.6998166830064
x5=91.8352138201382x_{5} = 91.8352138201382
x6=64.2538330751926x_{6} = 64.2538330751926
x7=99.7694765326183x_{7} = 99.7694765326183
x8=105.728201054821x_{8} = 105.728201054821
x9=62.3051907182457x_{9} = 62.3051907182457
x10=56.4939044922759x_{10} = 56.4939044922759
x11=121.641881995917x_{11} = 121.641881995917
x12=66.2069712419647x_{12} = 66.2069712419647
x13=81.9412486398235x_{13} = 81.9412486398235
x14=41.582662947005x_{14} = 41.582662947005
x15=97.7846464707569x_{15} = 97.7846464707569
x16=48.8745515352925x_{16} = 48.8745515352925
x17=43.352726865746x_{17} = 43.352726865746
x18=119.65116034046x_{18} = 119.65116034046
x19=107.715693682187x_{19} = 107.715693682187
x20=74.054379018256x_{20} = 74.054379018256
x21=58.4242921545119x_{21} = 58.4242921545119
x22=117.660817711865x_{22} = 117.660817711865
x23=50.7597542331829x_{23} = 50.7597542331829
x24=36.7920223210793x_{24} = 36.7920223210793
x25=95.8006149653489x_{25} = 95.8006149653489
x26=83.9173895835772x_{26} = 83.9173895835772
x27=76.0230615754436x_{27} = 76.0230615754436
x28=79.9666995493638x_{28} = 79.9666995493638
x29=111.692310969008x_{29} = 111.692310969008
x30=70.1245493821957x_{30} = 70.1245493821957
x31=72.088110991525x_{31} = 72.088110991525
x32=87.8738839811459x_{32} = 87.8738839811459
x33=113.681366227092x_{33} = 113.681366227092
x34=52.6597728585125x_{34} = 52.6597728585125
x35=60.3617313520525x_{35} = 60.3617313520525
x36=54.5718551293868x_{36} = 54.5718551293868
x37=103.741304214401x_{37} = 103.741304214401
x38=115.670877779771x_{38} = 115.670877779771
x39=77.9939075218715x_{39} = 77.9939075218715
x40=93.8174467631359x_{40} = 93.8174467631359
x41=45.1648087572857x_{41} = 45.1648087572857
x42=101.755046730513x_{42} = 101.755046730513
x43=89.8539963373609x_{43} = 89.8539963373609
x44=7.12503576604863x_{44} = 7.12503576604863
x45=47.0078639535934x_{45} = 47.0078639535934
x46=68.1640355218935x_{46} = 68.1640355218935
x47=39.8723677312301x_{47} = 39.8723677312301
x48=2.1983066219078x_{48} = 2.1983066219078
x49=38.253348001475x_{49} = 38.253348001475

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
[0.300644050042844,2.1983066219078]∪[7.12503576604863,∞)\left[0.300644050042844, 2.1983066219078\right] \cup \left[7.12503576604863, \infty\right)
Convexa en los intervalos
(−∞,0.300644050042844]∪[2.1983066219078,7.12503576604863]\left(-\infty, 0.300644050042844\right] \cup \left[2.1983066219078, 7.12503576604863\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x((−1)3x4+xlog⁡(x)2)e−x)=∞\lim_{x \to -\infty}\left(x \left(\frac{\left(-1\right) 3 x}{4} + \frac{x \log{\left(x \right)}}{2}\right) e^{- x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(x((−1)3x4+xlog⁡(x)2)e−x)=0\lim_{x \to \infty}\left(x \left(\frac{\left(-1\right) 3 x}{4} + \frac{x \log{\left(x \right)}}{2}\right) e^{- x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x*((-3*x)/4 + (x*log(x))/2))*exp(-x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((−1)3x4+xlog⁡(x)2)e−x)=−∞\lim_{x \to -\infty}\left(\left(\frac{\left(-1\right) 3 x}{4} + \frac{x \log{\left(x \right)}}{2}\right) e^{- x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞(((−1)3x4+xlog⁡(x)2)e−x)=0\lim_{x \to \infty}\left(\left(\frac{\left(-1\right) 3 x}{4} + \frac{x \log{\left(x \right)}}{2}\right) e^{- x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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:
x((−1)3x4+xlog⁡(x)2)e−x=−x(−xlog⁡(−x)2+3x4)exx \left(\frac{\left(-1\right) 3 x}{4} + \frac{x \log{\left(x \right)}}{2}\right) e^{- x} = - x \left(- \frac{x \log{\left(- x \right)}}{2} + \frac{3 x}{4}\right) e^{x}
- No
x((−1)3x4+xlog⁡(x)2)e−x=x(−xlog⁡(−x)2+3x4)exx \left(\frac{\left(-1\right) 3 x}{4} + \frac{x \log{\left(x \right)}}{2}\right) e^{- x} = x \left(- \frac{x \log{\left(- x \right)}}{2} + \frac{3 x}{4}\right) e^{x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = x*(-3*x/4+x*log(x)/2)*exp(-x)
    © Sr Examen – Калькулятор