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Plotting a function graph/ e^x*x^2

Graphing y = e^x*x^2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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