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

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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        3  -x
f(x) = x *E
f(x)=e−xx3f{\left(x \right)} = e^{- x} x^{3} 
f = E^(-x)*x^3
The graph of the function
02468-8-6-4-2-1010-2000000020000000
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} x^{3} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=113.817945104066x_{1} = 113.817945104066
x2=80.1510345473422x_{2} = 80.1510345473422
x3=47.2258221026002x_{3} = 47.2258221026002
x4=90.0216356011828x_{4} = 90.0216356011828
x5=105.873885239726x_{5} = 105.873885239726
x6=117.793236913112x_{6} = 117.793236913112
x7=66.4179766096377x_{7} = 66.4179766096377
x8=49.0998156927321x_{8} = 49.0998156927321
x9=74.2498293547747x_{9} = 74.2498293547747
x10=93.9790749415684x_{10} = 93.9790749415684
x11=86.0690060516037x_{11} = 86.0690060516037
x12=45.3699033599292x_{12} = 45.3699033599292
x13=78.181864782784x_{13} = 78.181864782784
x14=76.2147268831127x_{14} = 76.2147268831127
x15=72.2874103791773x_{15} = 72.2874103791773
x16=82.1220528473812x_{16} = 82.1220528473812
x17=101.905718658495x_{17} = 101.905718658495
x18=107.858996843108x_{18} = 107.858996843108
x19=54.8012720585185x_{19} = 54.8012720585185
x20=52.8897741765516x_{20} = 52.8897741765516
x21=99.9227607635738x_{21} = 99.9227607635738
x22=95.9593746156686x_{22} = 95.9593746156686
x23=109.844736107553x_{23} = 109.844736107553
x24=56.7215653754984x_{24} = 56.7215653754984
x25=97.9406256913241x_{25} = 97.9406256913241
x26=60.583728892351x_{26} = 60.583728892351
x27=103.889443728221x_{27} = 103.889443728221
x28=115.805346154896x_{28} = 115.805346154896
x29=50.9886343393585x_{29} = 50.9886343393585
x30=62.5237226565755x_{30} = 62.5237226565755
x31=64.4686693421837x_{31} = 64.4686693421837
x32=−9.61894480741186⋅10−5x_{32} = -9.61894480741186 \cdot 10^{-5}
x33=88.0446699300268x_{33} = 88.0446699300268
x34=119.7815893439x_{34} = 119.7815893439
x35=39.9621397880181x_{35} = 39.9621397880181
x36=41.7310513736826x_{36} = 41.7310513736826
x37=84.0947578295009x_{37} = 84.0947578295009
x38=91.9998011210345x_{38} = 91.9998011210345
x39=121.770377514453x_{39} = 121.770377514453
x40=68.3711434889037x_{40} = 68.3711434889037
x41=58.6493938015257x_{41} = 58.6493938015257
x42=0x_{42} = 0
x43=43.536364764524x_{43} = 43.536364764524
x44=111.831064115115x_{44} = 111.831064115115
x45=70.3277433163808x_{45} = 70.3277433163808
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^3*E^(-x).
03e−00^{3} 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+3x2e−x=0- x^{3} e^{- x} + 3 x^{2} e^{- x} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=3x_{2} = 3
The values of the extrema at the points:
(0, 0)

        -3 
(3, 27*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:
The function has no minima
Maxima of the function at points:
x2=3x_{2} = 3
Decreasing at intervals
(−∞,3]\left(-\infty, 3\right]
Increasing at intervals
[3,∞)\left[3, \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
x(x2−6x+6)e−x=0x \left(x^{2} - 6 x + 6\right) e^{- x} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=3−3x_{2} = 3 - \sqrt{3}
x3=3+3x_{3} = \sqrt{3} + 3

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
[0,3−3]∪[3+3,∞)\left[0, 3 - \sqrt{3}\right] \cup \left[\sqrt{3} + 3, \infty\right)
Convex at the intervals
(−∞,0]∪[3−3,3+3]\left(-\infty, 0\right] \cup \left[3 - \sqrt{3}, \sqrt{3} + 3\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)=−∞\lim_{x \to -\infty}\left(e^{- x} x^{3}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(e−xx3)=0\lim_{x \to \infty}\left(e^{- x} x^{3}\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^3*E^(-x), divided by x 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,
inclined 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,
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} x^{3} = - x^{3} e^{x}
- No
e−xx3=x3exe^{- x} x^{3} = x^{3} e^{x}
- No
so, the function
not is
neither even, nor odd
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