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

Graphing y = x*e^(2x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2*x
f(x) = x*E
f(x)=e2xxf{\left(x \right)} = e^{2 x} x 
f = E^(2*x)*x
The graph of the function
02468-8-6-4-2-1010-50000000005000000000
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:
e2xx=0e^{2 x} x = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−62.4594813057761x_{1} = -62.4594813057761
x2=−22.6957023751319x_{2} = -22.6957023751319
x3=−96.4242823853152x_{3} = -96.4242823853152
x4=−48.4905367883253x_{4} = -48.4905367883253
x5=−74.443042965628x_{5} = -74.443042965628
x6=−58.4666463153532x_{6} = -58.4666463153532
x7=−88.4299412042358x_{7} = -88.4299412042358
x8=−108.417456216542x_{8} = -108.417456216542
x9=−44.5036237757639x_{9} = -44.5036237757639
x10=−60.4629310925067x_{10} = -60.4629310925067
x11=−98.4230212294978x_{11} = -98.4230212294978
x12=−100.421813657552x_{12} = -100.421813657552
x13=−104.419546152707x_{13} = -104.419546152707
x14=−66.4532716391802x_{14} = -66.4532716391802
x15=−18.8120890441258x_{15} = -18.8120890441258
x16=−84.4332052360421x_{16} = -84.4332052360421
x17=−15.0740840979127x_{17} = -15.0740840979127
x18=−28.6042039159275x_{18} = -28.6042039159275
x19=−78.4387803330419x_{19} = -78.4387803330419
x20=−80.4368216405647x_{20} = -80.4368216405647
x21=−76.4408508191288x_{21} = -76.4408508191288
x22=−64.4562694336153x_{22} = -64.4562694336153
x23=−30.5841669729212x_{23} = -30.5841669729212
x24=−54.4750062227357x_{24} = -54.4750062227357
x25=−92.4269803908933x_{25} = -92.4269803908933
x26=−20.7448218335939x_{26} = -20.7448218335939
x27=−56.4706589232168x_{27} = -56.4706589232168
x28=−72.4453678375428x_{28} = -72.4453678375428
x29=−16.9108476139709x_{29} = -16.9108476139709
x30=−40.5198064107757x_{30} = -40.5198064107757
x31=−70.4478378859715x_{31} = -70.4478378859715
x32=−82.434965914994x_{32} = -82.434965914994
x33=−90.4284256014174x_{33} = -90.4284256014174
x34=−68.4504671725702x_{34} = -68.4504671725702
x35=0x_{35} = 0
x36=−94.4256007756744x_{36} = -94.4256007756744
x37=−52.479732054378x_{37} = -52.479732054378
x38=−34.5528319076254x_{38} = -34.5528319076254
x39=−38.5294259176999x_{39} = -38.5294259176999
x40=−24.6581187031698x_{40} = -24.6581187031698
x41=−110.416471439679x_{41} = -110.416471439679
x42=−42.5112629711588x_{42} = -42.5112629711588
x43=−46.4967518857691x_{43} = -46.4967518857691
x44=−102.420656323043x_{44} = -102.420656323043
x45=−26.628369572651x_{45} = -26.628369572651
x46=−106.418480319111x_{46} = -106.418480319111
x47=−36.5403401551302x_{47} = -36.5403401551302
x48=−50.4848882937228x_{48} = -50.4848882937228
x49=−32.567273706796x_{49} = -32.567273706796
x50=−86.4315324762772x_{50} = -86.4315324762772
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*E^(2*x).
0e0⋅20 e^{0 \cdot 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
2xe2x+e2x=02 x e^{2 x} + e^{2 x} = 0
Solve this equation
The roots of this equation
x1=−12x_{1} = - \frac{1}{2}
The values of the extrema at the points:
         -1  
       -e    
(-1/2, -----)
         2


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=−12x_{1} = - \frac{1}{2}
The function has no maxima
Decreasing at intervals
[−12,∞)\left[- \frac{1}{2}, \infty\right)
Increasing at intervals
(−∞,−12]\left(-\infty, - \frac{1}{2}\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
4(x+1)e2x=04 \left(x + 1\right) e^{2 x} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1

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
[−1,∞)\left[-1, \infty\right)
Convex at the intervals
(−∞,−1]\left(-\infty, -1\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(e2xx)=0\lim_{x \to -\infty}\left(e^{2 x} x\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(e2xx)=∞\lim_{x \to \infty}\left(e^{2 x} x\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*E^(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞e2x=0\lim_{x \to -\infty} e^{2 x} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞e2x=∞\lim_{x \to \infty} e^{2 x} = \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:
e2xx=−xe−2xe^{2 x} x = - x e^{- 2 x}
- No
e2xx=xe−2xe^{2 x} x = x e^{- 2 x}
- No
so, the function
not is
neither even, nor odd
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