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Plotting a function graph/ 2xexp^(-x/2)

Graphing y = 2xexp^(-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) = 2*x*E
f(x)=e(−1)x2⋅2xf{\left(x \right)} = e^{\frac{\left(-1\right) x}{2}} \cdot 2 x 
f = E^((-x)/2)*(2*x)
The graph of the function
02468-8-6-4-2-1010-50005000
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)x2⋅2x=0e^{\frac{\left(-1\right) x}{2}} \cdot 2 x = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=117.125411922138x_{1} = 117.125411922138
x2=75.9582278615682x_{2} = 75.9582278615682
x3=123.067540388527x_{3} = 123.067540388527
x4=89.5484716110773x_{4} = 89.5484716110773
x5=113.168557011776x_{5} = 113.168557011776
x6=85.6439180738776x_{6} = 85.6439180738776
x7=87.5945090232618x_{7} = 87.5945090232618
x8=72.1286573308603x_{8} = 72.1286573308603
x9=77.8843596511898x_{9} = 77.8843596511898
x10=115.146485250814x_{10} = 115.146485250814
x11=107.241540269193x_{11} = 107.241540269193
x12=111.191701047147x_{12} = 111.191701047147
x13=121.08599800789x_{13} = 121.08599800789
x14=0x_{14} = 0
x15=81.7545134822841x_{15} = 81.7545134822841
x16=125.04984591054x_{16} = 125.04984591054
x17=95.427382421153x_{17} = 95.427382421153
x18=77.5601992651609x_{18} = 77.5601992651609
x19=79.816716308387x_{19} = 79.816716308387
x20=131.000891064693x_{20} = 131.000891064693
x21=142.918298209055x_{21} = 142.918298209055
x22=119.105269897573x_{22} = 119.105269897573
x23=138.943848589893x_{23} = 138.943848589893
x24=93.4651859652441x_{24} = 93.4651859652441
x25=91.5054628829366x_{25} = 91.5054628829366
x26=99.3583181793708x_{26} = 99.3583181793708
x27=101.326683040058x_{27} = 101.326683040058
x28=74.0392717219567x_{28} = 74.0392717219567
x29=64.605232251426x_{29} = 64.605232251426
x30=70.2278341185476x_{30} = 70.2278341185476
x31=134.971304934036x_{31} = 134.971304934036
x32=129.016562623174x_{32} = 129.016562623174
x33=105.268425321898x_{33} = 105.268425321898
x34=68.3386317231503x_{34} = 68.3386317231503
x35=109.215998787545x_{35} = 109.215998787545
x36=132.985816431156x_{36} = 132.985816431156
x37=140.930847885457x_{37} = 140.930847885457
x38=83.6970973965431x_{38} = 83.6970973965431
x39=103.296764962881x_{39} = 103.296764962881
x40=66.4634017838308x_{40} = 66.4634017838308
x41=97.3918261051326x_{41} = 97.3918261051326
x42=136.957325310529x_{42} = 136.957325310529
x43=127.032867683997x_{43} = 127.032867683997
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (2*x)*E^((-x)/2).
0⋅2e(−1)020 \cdot 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
−xe(−1)x2+2e(−1)x2=0- x e^{\frac{\left(-1\right) x}{2}} + 2 e^{\frac{\left(-1\right) x}{2}} = 0
Solve this equation
The roots of this equation
x1=2x_{1} = 2
The values of the extrema at the points:
       -1 
(2, 4*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:
x1=2x_{1} = 2
Decreasing at intervals
(−∞,2]\left(-\infty, 2\right]
Increasing at intervals
[2,∞)\left[2, \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
(x2−2)e−x2=0\left(\frac{x}{2} - 2\right) e^{- \frac{x}{2}} = 0
Solve this equation
The roots of this equation
x1=4x_{1} = 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,∞)\left[4, \infty\right)
Convex at the intervals
(−∞,4]\left(-\infty, 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)x2⋅2x)=−∞\lim_{x \to -\infty}\left(e^{\frac{\left(-1\right) x}{2}} \cdot 2 x\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(e(−1)x2⋅2x)=0\lim_{x \to \infty}\left(e^{\frac{\left(-1\right) x}{2}} \cdot 2 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 (2*x)*E^((-x)/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2e(−1)x2)=∞\lim_{x \to -\infty}\left(2 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→∞(2e(−1)x2)=0\lim_{x \to \infty}\left(2 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)x2⋅2x=−2xex2e^{\frac{\left(-1\right) x}{2}} \cdot 2 x = - 2 x e^{\frac{x}{2}}
- No
e(−1)x2⋅2x=2xex2e^{\frac{\left(-1\right) x}{2}} \cdot 2 x = 2 x e^{\frac{x}{2}}
- No
so, the function
not is
neither even, nor odd
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