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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)=exx3f{\left(x \right)} = e^{x} x^{3} 
f = E^x*x^3
The graph of the function
02468-8-6-4-2-1010-2500000025000000
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:
exx3=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=−81.8167544117873x_{1} = -81.8167544117873
x2=−43.254793289805x_{2} = -43.254793289805
x3=−77.8771426363418x_{3} = -77.8771426363418
x4=−54.5046825561654x_{4} = -54.5046825561654
x5=−105.566581155148x_{5} = -105.566581155148
x6=−62.2229958168436x_{6} = -62.2229958168436
x7=−119.473696806211x_{7} = -119.473696806211
x8=−71.9837938598591x_{8} = -71.9837938598591
x9=−103.58224722089x_{9} = -103.58224722089
x10=−85.7632268036374x_{10} = -85.7632268036374
x11=−48.8085971699827x_{11} = -48.8085971699827
x12=−56.4237044386907x_{12} = -56.4237044386907
x13=−68.068485074228x_{13} = -68.068485074228
x14=−39.6921638743108x_{14} = -39.6921638743108
x15=−60.2838279161017x_{15} = -60.2838279161017
x16=−87.7386796067909x_{16} = -87.7386796067909
x17=−101.598637273947x_{17} = -101.598637273947
x18=0x_{18} = 0
x19=−115.49759696096x_{19} = -115.49759696096
x20=−6.37672685750786⋅10−5x_{20} = -6.37672685750786 \cdot 10^{-5}
x21=−89.7154509915966x_{21} = -89.7154509915966
x22=−79.846010822632x_{22} = -79.846010822632
x23=−73.9458061467892x_{23} = -73.9458061467892
x24=−91.6934372760935x_{24} = -91.6934372760935
x25=−113.510274213085x_{25} = -113.510274213085
x26=−97.6338001722932x_{26} = -97.6338001722932
x27=−83.7892084427348x_{27} = -83.7892084427348
x28=−66.1158854871866x_{28} = -66.1158854871866
x29=−52.5946760133184x_{29} = -52.5946760133184
x30=−99.615802770923x_{30} = -99.615802770923
x31=−117.485413951559x_{31} = -117.485413951559
x32=−95.6526915671241x_{32} = -95.6526915671241
x33=−50.6953021085607x_{33} = -50.6953021085607
x34=−109.537236988787x_{34} = -109.537236988787
x35=−93.6725453940216x_{35} = -93.6725453940216
x36=−121.46241928026x_{36} = -121.46241928026
x37=−111.523476442329x_{37} = -111.523476442329
x38=−41.454503250211x_{38} = -41.454503250211
x39=−70.0245793288288x_{39} = -70.0245793288288
x40=−107.551592080799x_{40} = -107.551592080799
x41=−58.3504397456909x_{41} = -58.3504397456909
x42=−45.0843950117395x_{42} = -45.0843950117395
x43=−75.9103368174603x_{43} = -75.9103368174603
x44=−64.1672177737049x_{44} = -64.1672177737049
x45=−46.9371649842841x_{45} = -46.9371649842841
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.
03e00^{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
x3ex+3x2ex=0x^{3} e^{x} + 3 x^{2} e^{x} = 0
Solve this equation
The roots of this equation
x1=−3x_{1} = -3
x2=0x_{2} = 0
The values of the extrema at the points:
          -3 
(-3, -27*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=−3x_{1} = -3
The function has no maxima
Decreasing at intervals
[−3,∞)\left[-3, \infty\right)
Increasing at intervals
(−∞,−3]\left(-\infty, -3\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)ex=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} = -3 + \sqrt{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
[−3−3,−3+3]∪[0,∞)\left[-3 - \sqrt{3}, -3 + \sqrt{3}\right] \cup \left[0, \infty\right)
Convex at the intervals
(−∞,−3−3]∪[−3+3,0]\left(-\infty, -3 - \sqrt{3}\right] \cup \left[-3 + \sqrt{3}, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(exx3)=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 left:
y=0y = 0
lim⁡x→∞(exx3)=∞\lim_{x \to \infty}\left(e^{x} x^{3}\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^3*E^x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x2ex)=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 right
lim⁡x→∞(x2ex)=∞\lim_{x \to \infty}\left(x^{2} 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:
exx3=−x3e−xe^{x} x^{3} = - x^{3} e^{- x}
- No
exx3=x3e−xe^{x} x^{3} = x^{3} e^{- x}
- No
so, the function
not is
neither even, nor odd
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