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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

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        2  x
f(x) = x *E
f(x)=exx2f{\left(x \right)} = e^{x} x^{2} 
f = E^x*x^2
The graph of the function
02468-8-6-4-2-101002500000
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:
exx2=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=−89.4277891533326x_{1} = -89.4277891533326
x2=−40.5471004173384x_{2} = -40.5471004173384
x3=−69.628833400408x_{3} = -69.628833400408
x4=−46.2166624604922x_{4} = -46.2166624604922
x5=−61.757295261576x_{5} = -61.757295261576
x6=0x_{6} = 0
x7=−109.31131787361x_{7} = -109.31131787361
x8=−79.5128437462747x_{8} = -79.5128437462747
x9=−36.8813855334114x_{9} = -36.8813855334114
x10=−63.7212246430644x_{10} = -63.7212246430644
x11=−111.302305760974x_{11} = -111.302305760974
x12=−91.413426044512x_{12} = -91.413426044512
x13=−105.330526752392x_{13} = -105.330526752392
x14=−53.9389966224242x_{14} = -53.9389966224242
x15=−81.4938033513721x_{15} = -81.4938033513721
x16=−97.3744818786337x_{16} = -97.3744818786337
x17=−107.320716385987x_{17} = -107.320716385987
x18=−119.269680169774x_{18} = -119.269680169774
x19=−57.8395946559803x_{19} = -57.8395946559803
x20=−93.3997888155798x_{20} = -93.3997888155798
x21=−101.351496587439x_{21} = -101.351496587439
x22=−35.1082010514801x_{22} = -35.1082010514801
x23=−55.886836936279x_{23} = -55.886836936279
x24=−117.277362966189x_{24} = -117.277362966189
x25=−65.6880004393027x_{25} = -65.6880004393027
x26=−67.6572960646381x_{26} = -67.6572960646381
x27=−42.4197387542301x_{27} = -42.4197387542301
x28=−113.293656653183x_{28} = -113.293656653183
x29=−115.285349010188x_{29} = -115.285349010188
x30=−75.5546705895527x_{30} = -75.5546705895527
x31=−38.6983611853733x_{31} = -38.6983611853733
x32=−51.9968968445388x_{32} = -51.9968968445388
x33=−87.4429379040025x_{33} = -87.4429379040025
x34=−59.7965985080519x_{34} = -59.7965985080519
x35=−121.262283642069x_{35} = -121.262283642069
x36=−99.3627195189532x_{36} = -99.3627195189532
x37=−44.3108762649905x_{37} = -44.3108762649905
x38=−85.4589388313701x_{38} = -85.4589388313701
x39=−95.3868236343622x_{39} = -95.3868236343622
x40=−83.4758662349933x_{40} = -83.4758662349933
x41=−77.5330929772024x_{41} = -77.5330929772024
x42=−103.340776718801x_{42} = -103.340776718801
x43=−48.134267415089x_{43} = -48.134267415089
x44=−73.5777125278413x_{44} = -73.5777125278413
x45=−50.061558962287x_{45} = -50.061558962287
x46=−71.6023740669893x_{46} = -71.6023740669893
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.
02e00^{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
x2ex+2xex=0x^{2} e^{x} + 2 x e^{x} = 0
Solve this equation
The roots of this equation
x1=−2x_{1} = -2
x2=0x_{2} = 0
The values of the extrema at the points:
        -2 
(-2, 4*e  )

(0, 0)


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
(−∞,−2]∪[0,∞)\left(-\infty, -2\right] \cup \left[0, \infty\right)
Increasing at intervals
[−2,0]\left[-2, 0\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)ex=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} = -2 + \sqrt{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[-2 + \sqrt{2}, \infty\right)
Convex at the intervals
[−2−2,−2+2]\left[-2 - \sqrt{2}, -2 + \sqrt{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→−∞(exx2)=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 left:
y=0y = 0
lim⁡x→∞(exx2)=∞\lim_{x \to \infty}\left(e^{x} x^{2}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
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→−∞(xex)=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 right
lim⁡x→∞(xex)=∞\lim_{x \to \infty}\left(x e^{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
exx2=x2e−xe^{x} x^{2} = x^{2} e^{- x}
- No
exx2=−x2e−xe^{x} x^{2} = - x^{2} e^{- x}
- No
so, the function
not is
neither even, nor odd
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