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Plotting a function graph/ x^2*e^(-x)

Graphing y = x^2*e^(-x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        2  -x
f(x) = x *E
f(x)=e−xx2f{\left(x \right)} = e^{- x} x^{2} 
f = E^(-x)*x^2
The graph of the function
1.002.001.101.201.301.401.501.601.701.801.900.20.6
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−xx2=0e^{- x} x^{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=99.6706130283057x_{1} = 99.6706130283057
x2=91.720934730719x_{2} = 91.720934730719
x3=121.570827102163x_{3} = 121.570827102163
x4=101.659470122749x_{4} = 101.659470122749
x5=115.593756384128x_{5} = 115.593756384128
x6=81.8006238116621x_{6} = 81.8006238116621
x7=40.8356618339334x_{7} = 40.8356618339334
x8=50.3607330233137x_{8} = 50.3607330233137
x9=0x_{9} = 0
x10=46.5128714785856x_{10} = 46.5128714785856
x11=35.379255492682x_{11} = 35.379255492682
x12=64.0255739002577x_{12} = 64.0255739002577
x13=58.1423474863896x_{13} = 58.1423474863896
x14=38.9827879874711x_{14} = 38.9827879874711
x15=77.8395419968606x_{15} = 77.8395419968606
x16=85.7660696193442x_{16} = 85.7660696193442
x17=75.8609058011359x_{17} = 75.8609058011359
x18=42.7114678029016x_{18} = 42.7114678029016
x19=40.5820728530031x_{19} = 40.5820728530031
x20=56.1888924840258x_{20} = 56.1888924840258
x21=103.648824952827x_{21} = 103.648824952827
x22=83.7828486140689x_{22} = 83.7828486140689
x23=71.9081118282112x_{23} = 71.9081118282112
x24=62.0611807434853x_{24} = 62.0611807434853
x25=73.8837117221529x_{25} = 73.8837117221529
x26=44.6050925906729x_{26} = 44.6050925906729
x27=93.707404744577x_{27} = 93.707404744577
x28=117.585818346237x_{28} = 117.585818346237
x29=52.2971932633301x_{29} = 52.2971932633301
x30=67.9624187188197x_{30} = 67.9624187188197
x31=111.610608082484x_{31} = 111.610608082484
x32=119.578180845004x_{32} = 119.578180845004
x33=109.619562634492x_{33} = 109.619562634492
x34=69.9342805013838x_{34} = 69.9342805013838
x35=105.638644821409x_{35} = 105.638644821409
x36=87.7502050583631x_{36} = 87.7502050583631
x37=37.1602455397125x_{37} = 37.1602455397125
x38=89.7351819043081x_{38} = 89.7351819043081
x39=107.628899840344x_{39} = 107.628899840344
x40=113.602013088993x_{40} = 113.602013088993
x41=97.6822895145426x_{41} = 97.6822895145426
x42=54.2402420845623x_{42} = 54.2402420845623
x43=65.9927593677372x_{43} = 65.9927593677372
x44=79.8194870788507x_{44} = 79.8194870788507
x45=60.0999560358985x_{45} = 60.0999560358985
x46=48.4320998819442x_{46} = 48.4320998819442
x47=95.6945389638031x_{47} = 95.6945389638031
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^2*E^(-x).
02e−00^{2} 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
−x2e−x+2xe−x=0- x^{2} e^{- x} + 2 x e^{- x} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=2x_{2} = 2
The values of the extrema at the points:
(0, 0)

       -2 
(2, 4*e  )


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:
Minima of the function at points:
x1=0x_{1} = 0
Maxima of the function at points:
x1=2x_{1} = 2
Decreasing at intervals
[0,2]\left[0, 2\right]
Increasing at intervals
(−∞,0]∪[2,∞)\left(-\infty, 0\right] \cup \left[2, \infty\right)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
(x2−4x+2)e−x=0\left(x^{2} - 4 x + 2\right) e^{- x} = 0
Solve this equation
The roots of this equation
x1=2−2x_{1} = 2 - \sqrt{2}
x2=2+2x_{2} = \sqrt{2} + 2

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,2−2]∪[2+2,∞)\left(-\infty, 2 - \sqrt{2}\right] \cup \left[\sqrt{2} + 2, \infty\right)
Convex at the intervals
[2−2,2+2]\left[2 - \sqrt{2}, \sqrt{2} + 2\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−xx2)=∞\lim_{x \to -\infty}\left(e^{- x} x^{2}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(e−xx2)=0\lim_{x \to \infty}\left(e^{- x} x^{2}\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^2*E^(-x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(xe−x)=−∞\lim_{x \to -\infty}\left(x e^{- x}\right) = -\infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞(xe−x)=0\lim_{x \to \infty}\left(x e^{- x}\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−xx2=x2exe^{- x} x^{2} = x^{2} e^{x}
- No
e−xx2=−x2exe^{- x} x^{2} = - x^{2} e^{x}
- No
so, the function
not is
neither even, nor odd
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