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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)=x3exf{\left(x \right)} = \frac{x^{3}}{e^{x}} 
f = x^3/E^x
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:
x3ex=0\frac{x^{3}}{e^{x}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=68.3711434889037x_{1} = 68.3711434889037
x2=86.0690060516037x_{2} = 86.0690060516037
x3=119.7815893439x_{3} = 119.7815893439
x4=91.9998011210345x_{4} = 91.9998011210345
x5=121.770377514453x_{5} = 121.770377514453
x6=64.4686693421837x_{6} = 64.4686693421837
x7=101.905718658495x_{7} = 101.905718658495
x8=58.6493938015257x_{8} = 58.6493938015257
x9=93.9790749415684x_{9} = 93.9790749415684
x10=74.2498293547747x_{10} = 74.2498293547747
x11=39.9621397880181x_{11} = 39.9621397880181
x12=113.817945104066x_{12} = 113.817945104066
x13=78.181864782784x_{13} = 78.181864782784
x14=109.844736107553x_{14} = 109.844736107553
x15=88.0446699300268x_{15} = 88.0446699300268
x16=47.2258221026002x_{16} = 47.2258221026002
x17=107.858996843108x_{17} = 107.858996843108
x18=72.2874103791773x_{18} = 72.2874103791773
x19=56.7215653754984x_{19} = 56.7215653754984
x20=97.9406256913241x_{20} = 97.9406256913241
x21=105.873885239726x_{21} = 105.873885239726
x22=80.1510345473422x_{22} = 80.1510345473422
x23=52.8897741765516x_{23} = 52.8897741765516
x24=41.7310513736826x_{24} = 41.7310513736826
x25=95.9593746156686x_{25} = 95.9593746156686
x26=43.536364764524x_{26} = 43.536364764524
x27=50.9886343393585x_{27} = 50.9886343393585
x28=115.805346154896x_{28} = 115.805346154896
x29=84.0947578295009x_{29} = 84.0947578295009
x30=70.3277433163808x_{30} = 70.3277433163808
x31=90.0216356011828x_{31} = 90.0216356011828
x32=82.1220528473812x_{32} = 82.1220528473812
x33=66.4179766096377x_{33} = 66.4179766096377
x34=0x_{34} = 0
x35=54.8012720585185x_{35} = 54.8012720585185
x36=99.9227607635738x_{36} = 99.9227607635738
x37=45.3699033599292x_{37} = 45.3699033599292
x38=62.5237226565755x_{38} = 62.5237226565755
x39=111.831064115115x_{39} = 111.831064115115
x40=117.793236913112x_{40} = 117.793236913112
x41=76.2147268831127x_{41} = 76.2147268831127
x42=49.0998156927321x_{42} = 49.0998156927321
x43=103.889443728221x_{43} = 103.889443728221
x44=−9.61894480741186⋅10−5x_{44} = -9.61894480741186 \cdot 10^{-5}
x45=60.583728892351x_{45} = 60.583728892351
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.
03e0\frac{0^{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→−∞(x3ex)=−∞\lim_{x \to -\infty}\left(\frac{x^{3}}{e^{x}}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x3ex)=0\lim_{x \to \infty}\left(\frac{x^{3}}{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^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:
x3ex=−x3ex\frac{x^{3}}{e^{x}} = - x^{3} e^{x}
- No
x3ex=x3ex\frac{x^{3}}{e^{x}} = x^{3} e^{x}
- No
so, the function
not is
neither even, nor odd
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