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  • Graphing y =:
  • (x^3-4x^2)(x^2-7)
  • x^3-2x^2-x
  • x^3-12x^2+145
  • sqrt(2x-1)-x
  • Integral of d{x}:
  • x^2*e^((-x)/2) x^2*e^((-x)/2)

  • Identical expressions

  • x^ two *e^((-x)/ two)
  • x squared multiply by e to the power of (( minus x) divide by 2)
  • x to the power of two multiply by e to the power of (( minus x) divide by two)
  • x2*e((-x)/2)
  • x2*e-x/2
  • x²*e^((-x)/2)
  • x to the power of 2*e to the power of ((-x)/2)
  • x^2e^((-x)/2)
  • x2e((-x)/2)
  • x2e-x/2
  • x^2e^-x/2
  • x^2*e^((-x) divide by 2)

  • Similar expressions

  • x^2*e^((x)/2)
Plotting a function graph/ x^2*e^((-x)/2)

Graphing y = x^2*e^((-x)/2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           -x 
           ---
        2   2 
f(x) = x *E
f(x)=e(−1)x2x2f{\left(x \right)} = e^{\frac{\left(-1\right) x}{2}} x^{2} 
f = E^((-x)/2)*x^2
The graph of the function
02468-8-6-4-2-1010020000
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(−1)x2x2=0e^{\frac{\left(-1\right) x}{2}} x^{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=122.001636542925x_{1} = 122.001636542925
x2=83.3907094556794x_{2} = 83.3907094556794
x3=92.8711313296711x_{3} = 92.8711313296711
x4=120.041406599039x_{4} = 120.041406599039
x5=111.903844163895x_{5} = 111.903844163895
x6=0x_{6} = 0
x7=89.0566471878038x_{7} = 89.0566471878038
x8=137.736603530702x_{8} = 137.736603530702
x9=72.3837747282901x_{9} = 72.3837747282901
x10=123.963564638001x_{10} = 123.963564638001
x11=143.656397501743x_{11} = 143.656397501743
x12=139.708897233921x_{12} = 139.708897233921
x13=100.565248683702x_{13} = 100.565248683702
x14=112.219995158752x_{14} = 112.219995158752
x15=74.1751207498496x_{15} = 74.1751207498496
x16=90.9606974407717x_{16} = 90.9606974407717
x17=102.499602121155x_{17} = 102.499602121155
x18=127.892094344434x_{18} = 127.892094344434
x19=79.6624387990361x_{19} = 79.6624387990361
x20=141.682179321777x_{20} = 141.682179321777
x21=81.5208130407554x_{21} = 81.5208130407554
x22=116.126522362797x_{22} = 116.126522362797
x23=98.6348274721793x_{23} = 98.6348274721793
x24=108.323137859362x_{24} = 108.323137859362
x25=135.765354503655x_{25} = 135.765354503655
x26=133.795210836192x_{26} = 133.795210836192
x27=106.378824113363x_{27} = 106.378824113363
x28=94.7873184046098x_{28} = 94.7873184046098
x29=77.8172635150345x_{29} = 77.8172635150345
x30=96.7087102837885x_{30} = 96.7087102837885
x31=104.437558301726x_{31} = 104.437558301726
x32=87.1597095047415x_{32} = 87.1597095047415
x33=129.858506970676x_{33} = 129.858506970676
x34=85.2707324526338x_{34} = 85.2707324526338
x35=125.927083409265x_{35} = 125.927083409265
x36=118.082992299051x_{36} = 118.082992299051
x37=110.270265102252x_{37} = 110.270265102252
x38=75.9873233164275x_{38} = 75.9873233164275
x39=114.172138128465x_{39} = 114.172138128465
x40=131.826238044109x_{40} = 131.826238044109
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)/2).
02e(−1)020^{2} e^{\frac{\left(-1\right) 0}{2}}
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
−x2e(−1)x22+2xe(−1)x2=0- \frac{x^{2} e^{\frac{\left(-1\right) x}{2}}}{2} + 2 x e^{\frac{\left(-1\right) x}{2}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=4x_{2} = 4
The values of the extrema at the points:
(0, 0)

        -2 
(4, 16*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:
Minima of the function at points:
x1=0x_{1} = 0
Maxima of the function at points:
x1=4x_{1} = 4
Decreasing at intervals
[0,4]\left[0, 4\right]
Increasing at intervals
(−∞,0]∪[4,∞)\left(-\infty, 0\right] \cup \left[4, \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
(x24−2x+2)e−x2=0\left(\frac{x^{2}}{4} - 2 x + 2\right) e^{- \frac{x}{2}} = 0
Solve this equation
The roots of this equation
x1=4−22x_{1} = 4 - 2 \sqrt{2}
x2=22+4x_{2} = 2 \sqrt{2} + 4

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
(−∞,4−22]∪[22+4,∞)\left(-\infty, 4 - 2 \sqrt{2}\right] \cup \left[2 \sqrt{2} + 4, \infty\right)
Convex at the intervals
[4−22,22+4]\left[4 - 2 \sqrt{2}, 2 \sqrt{2} + 4\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(−1)x2x2)=∞\lim_{x \to -\infty}\left(e^{\frac{\left(-1\right) x}{2}} x^{2}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(e(−1)x2x2)=0\lim_{x \to \infty}\left(e^{\frac{\left(-1\right) x}{2}} x^{2}\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^2*E^((-x)/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(xe(−1)x2)=−∞\lim_{x \to -\infty}\left(x e^{\frac{\left(-1\right) x}{2}}\right) = -\infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞(xe(−1)x2)=0\lim_{x \to \infty}\left(x e^{\frac{\left(-1\right) x}{2}}\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(−1)x2x2=x2ex2e^{\frac{\left(-1\right) x}{2}} x^{2} = x^{2} e^{\frac{x}{2}}
- No
e(−1)x2x2=−x2ex2e^{\frac{\left(-1\right) x}{2}} x^{2} = - x^{2} e^{\frac{x}{2}}
- No
so, the function
not is
neither even, nor odd
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