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  • How to use it?

  • Graphing y =:
  • -3x+5
  • 3x^4-4x^3
  • 3/2x^2-x^3
  • 3x^2-12x
  • Derivative of:
  • x^(1/3)*e^(-x) x^(1/3)*e^(-x)

  • Identical expressions

  • x^(one / three)*e^(-x)
  • x to the power of (1 divide by 3) multiply by e to the power of ( minus x)
  • x to the power of (one divide by three) multiply by e to the power of ( minus x)
  • x(1/3)*e(-x)
  • x1/3*e-x
  • x^(1/3)e^(-x)
  • x(1/3)e(-x)
  • x1/3e-x
  • x^1/3e^-x
  • x^(1 divide by 3)*e^(-x)

  • Similar expressions

  • x^(1/3)*e^(x)
Plotting a function graph/ x^(1/3)*e^(-x)

Graphing y = x^(1/3)*e^(-x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       3 ___  -x
f(x) = \/ x *E
f(x)=e−xx3f{\left(x \right)} = e^{- x} \sqrt[3]{x} 
f = E^(-x)*x^(1/3)
The graph of the function
02468-8-6-4-2-10100.01.0
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
e−xx3=0e^{- x} \sqrt[3]{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=33.5286981570306x_{1} = 33.5286981570306
x2=29.6269095273327x_{2} = 29.6269095273327
x3=39.4452332453898x_{3} = 39.4452332453898
x4=65.3133253157007x_{4} = 65.3133253157007
x5=111.252069207491x_{5} = 111.252069207491
x6=93.2676977460751x_{6} = 93.2676977460751
x7=89.2721674640506x_{7} = 89.2721674640506
x8=81.2826726466621x_{8} = 81.2826726466621
x9=119.246814492279x_{9} = 119.246814492279
x10=45.3951328784198x_{10} = 45.3951328784198
x11=35.4951987253783x_{11} = 35.4951987253783
x12=51.3613914384959x_{12} = 51.3613914384959
x13=75.2923193654619x_{13} = 75.2923193654619
x14=103.258249611654x_{14} = 103.258249611654
x15=105.256604588073x_{15} = 105.256604588073
x16=115.249339684993x_{16} = 115.249339684993
x17=79.2856943090103x_{17} = 79.2856943090103
x18=85.277130127622x_{18} = 85.277130127622
x19=121.245621461622x_{19} = 121.245621461622
x20=0x_{20} = 0
x21=91.2698757192509x_{21} = 91.2698757192509
x22=61.3241781954468x_{22} = 61.3241781954468
x23=57.3370136179432x_{23} = 57.3370136179432
x24=107.255029001296x_{24} = 107.255029001296
x25=97.2636506804946x_{25} = 97.2636506804946
x26=95.2656252377379x_{26} = 95.2656252377379
x27=113.250677380766x_{27} = 113.250677380766
x28=71.2998543086375x_{28} = 71.2998543086375
x29=49.3713562973095x_{29} = 49.3713562973095
x30=53.3524413315494x_{30} = 53.3524413315494
x31=55.3443558773275x_{31} = 55.3443558773275
x32=101.259968785345x_{32} = 101.259968785345
x33=99.2617672608143x_{33} = 99.2617672608143
x34=87.2745821883794x_{34} = 87.2745821883794
x35=83.279822693014x_{35} = 83.279822693014
x36=67.3085026225693x_{36} = 67.3085026225693
x37=77.288903806042x_{37} = 77.288903806042
x38=43.4094921706822x_{38} = 43.4094921706822
x39=109.253518530562x_{39} = 109.253518530562
x40=31.5710744304598x_{40} = 31.5710744304598
x41=47.3825238369606x_{41} = 47.3825238369606
x42=37.4679295763206x_{42} = 37.4679295763206
x43=73.2959616736576x_{43} = 73.2959616736576
x44=41.4260089466185x_{44} = 41.4260089466185
x45=69.3040242686795x_{45} = 69.3040242686795
x46=59.3303151318917x_{46} = 59.3303151318917
x47=117.248053013168x_{47} = 117.248053013168
x48=63.3185341779381x_{48} = 63.3185341779381
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^(1/3)*E^(-x).
03e−0\sqrt[3]{0} e^{- 0}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−x3e−x+e−x3x23=0- \sqrt[3]{x} e^{- x} + \frac{e^{- x}}{3 x^{\frac{2}{3}}} = 0
Solve this equation
The roots of this equation
x1=13x_{1} = \frac{1}{3}
The values of the extrema at the points:
       2/3  -1/3 
      3   *e     
(1/3, ----------)
          3


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x1=13x_{1} = \frac{1}{3}
Decreasing at intervals
(−∞,13]\left(-\infty, \frac{1}{3}\right]
Increasing at intervals
[13,∞)\left[\frac{1}{3}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(e−xx3)=∞sign⁡(−13)\lim_{x \to -\infty}\left(e^{- x} \sqrt[3]{x}\right) = \infty \operatorname{sign}{\left(\sqrt[3]{-1} \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=∞sign⁡(−13)y = \infty \operatorname{sign}{\left(\sqrt[3]{-1} \right)}
lim⁡x→∞(e−xx3)=0\lim_{x \to \infty}\left(e^{- x} \sqrt[3]{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x^(1/3)*E^(-x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(e−xx23)=−∞−13\lim_{x \to -\infty}\left(\frac{e^{- x}}{x^{\frac{2}{3}}}\right) = - \infty \sqrt[3]{-1}
Let's take the limit
so,
inclined asymptote equation on the left:
y=−∞−13xy = - \infty \sqrt[3]{-1} x
lim⁡x→∞(e−xx23)=0\lim_{x \to \infty}\left(\frac{e^{- x}}{x^{\frac{2}{3}}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
e−xx3=−x3exe^{- x} \sqrt[3]{x} = \sqrt[3]{- x} e^{x}
- No
e−xx3=−−x3exe^{- x} \sqrt[3]{x} = - \sqrt[3]{- x} e^{x}
- No
so, the function
not is
neither even, nor odd
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