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

Graphing y = x^2*exp(-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)=x2e−xf{\left(x \right)} = x^{2} e^{- x} 
f = x^2*exp(-x)
The graph of the function
02468-8-6-4-2-101002000000
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:
x2e−x=0x^{2} e^{- x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
x2=109.619562634492x_{2} = 109.619562634492
x3=79.8194870788507x_{3} = 79.8194870788507
x4=75.8609058011359x_{4} = 75.8609058011359
x5=95.6945389638031x_{5} = 95.6945389638031
x6=48.4320998819442x_{6} = 48.4320998819442
x7=69.9342805013838x_{7} = 69.9342805013838
x8=89.7351819043081x_{8} = 89.7351819043081
x9=81.8006238116621x_{9} = 81.8006238116621
x10=35.379255492682x_{10} = 35.379255492682
x11=87.7502050583631x_{11} = 87.7502050583631
x12=52.2971932633301x_{12} = 52.2971932633301
x13=105.638644821409x_{13} = 105.638644821409
x14=56.1888924840258x_{14} = 56.1888924840258
x15=58.1423474863896x_{15} = 58.1423474863896
x16=64.0255739002577x_{16} = 64.0255739002577
x17=44.6050925906729x_{17} = 44.6050925906729
x18=38.9827879874711x_{18} = 38.9827879874711
x19=85.7660696193442x_{19} = 85.7660696193442
x20=71.9081118282112x_{20} = 71.9081118282112
x21=67.9624187188197x_{21} = 67.9624187188197
x22=37.1602455397125x_{22} = 37.1602455397125
x23=60.0999560358985x_{23} = 60.0999560358985
x24=115.593756384128x_{24} = 115.593756384128
x25=121.570827102163x_{25} = 121.570827102163
x26=119.578180845004x_{26} = 119.578180845004
x27=113.602013088993x_{27} = 113.602013088993
x28=65.9927593677372x_{28} = 65.9927593677372
x29=91.720934730719x_{29} = 91.720934730719
x30=40.5820728530031x_{30} = 40.5820728530031
x31=97.6822895145426x_{31} = 97.6822895145426
x32=62.0611807434853x_{32} = 62.0611807434853
x33=50.3607330233137x_{33} = 50.3607330233137
x34=46.5128714785856x_{34} = 46.5128714785856
x35=77.8395419968606x_{35} = 77.8395419968606
x36=93.707404744577x_{36} = 93.707404744577
x37=101.659470122749x_{37} = 101.659470122749
x38=117.585818346237x_{38} = 117.585818346237
x39=40.8356618339334x_{39} = 40.8356618339334
x40=107.628899840344x_{40} = 107.628899840344
x41=99.6706130283057x_{41} = 99.6706130283057
x42=111.610608082484x_{42} = 111.610608082484
x43=42.7114678029016x_{43} = 42.7114678029016
x44=54.2402420845623x_{44} = 54.2402420845623
x45=103.648824952827x_{45} = 103.648824952827
x46=73.8837117221529x_{46} = 73.8837117221529
x47=83.7828486140689x_{47} = 83.7828486140689
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^2*exp(-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→−∞(x2e−x)=∞\lim_{x \to -\infty}\left(x^{2} e^{- x}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x2e−x)=0\lim_{x \to \infty}\left(x^{2} e^{- 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^2*exp(-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:
x2e−x=x2exx^{2} e^{- x} = x^{2} e^{x}
- No
x2e−x=−x2exx^{2} e^{- x} = - x^{2} e^{x}
- No
so, the function
not is
neither even, nor odd
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