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Plotting a function graph/ x*exp(2*x)

Graphing y = x*exp(2*x)

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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)=xe2xf{\left(x \right)} = x e^{2 x} 
f = x*exp(2*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:
xe2x=0x e^{2 x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−110.416471439679x_{1} = -110.416471439679
x2=−106.418480319111x_{2} = -106.418480319111
x3=−22.6957023751319x_{3} = -22.6957023751319
x4=−62.4594813057761x_{4} = -62.4594813057761
x5=−54.4750062227357x_{5} = -54.4750062227357
x6=−86.4315324762772x_{6} = -86.4315324762772
x7=−78.4387803330419x_{7} = -78.4387803330419
x8=−46.4967518857691x_{8} = -46.4967518857691
x9=−100.421813657552x_{9} = -100.421813657552
x10=−66.4532716391802x_{10} = -66.4532716391802
x11=0x_{11} = 0
x12=−76.4408508191288x_{12} = -76.4408508191288
x13=−44.5036237757639x_{13} = -44.5036237757639
x14=−42.5112629711588x_{14} = -42.5112629711588
x15=−94.4256007756744x_{15} = -94.4256007756744
x16=−40.5198064107757x_{16} = -40.5198064107757
x17=−18.8120890441258x_{17} = -18.8120890441258
x18=−48.4905367883253x_{18} = -48.4905367883253
x19=−90.4284256014174x_{19} = -90.4284256014174
x20=−60.4629310925067x_{20} = -60.4629310925067
x21=−80.4368216405647x_{21} = -80.4368216405647
x22=−16.9108476139709x_{22} = -16.9108476139709
x23=−82.434965914994x_{23} = -82.434965914994
x24=−92.4269803908933x_{24} = -92.4269803908933
x25=−20.7448218335939x_{25} = -20.7448218335939
x26=−24.6581187031698x_{26} = -24.6581187031698
x27=−96.4242823853152x_{27} = -96.4242823853152
x28=−104.419546152707x_{28} = -104.419546152707
x29=−38.5294259176999x_{29} = -38.5294259176999
x30=−26.628369572651x_{30} = -26.628369572651
x31=−50.4848882937228x_{31} = -50.4848882937228
x32=−30.5841669729212x_{32} = -30.5841669729212
x33=−15.0740840979127x_{33} = -15.0740840979127
x34=−58.4666463153532x_{34} = -58.4666463153532
x35=−28.6042039159275x_{35} = -28.6042039159275
x36=−70.4478378859715x_{36} = -70.4478378859715
x37=−102.420656323043x_{37} = -102.420656323043
x38=−34.5528319076254x_{38} = -34.5528319076254
x39=−72.4453678375428x_{39} = -72.4453678375428
x40=−98.4230212294978x_{40} = -98.4230212294978
x41=−68.4504671725702x_{41} = -68.4504671725702
x42=−32.567273706796x_{42} = -32.567273706796
x43=−36.5403401551302x_{43} = -36.5403401551302
x44=−52.479732054378x_{44} = -52.479732054378
x45=−84.4332052360421x_{45} = -84.4332052360421
x46=−74.443042965628x_{46} = -74.443042965628
x47=−64.4562694336153x_{47} = -64.4562694336153
x48=−56.4706589232168x_{48} = -56.4706589232168
x49=−108.417456216542x_{49} = -108.417456216542
x50=−88.4299412042358x_{50} = -88.4299412042358
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*exp(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→−∞(xe2x)=0\lim_{x \to -\infty}\left(x e^{2 x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(xe2x)=∞\lim_{x \to \infty}\left(x e^{2 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*exp(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:
xe2x=−xe−2xx e^{2 x} = - x e^{- 2 x}
- No
xe2x=xe−2xx e^{2 x} = x e^{- 2 x}
- No
so, the function
not is
neither even, nor odd
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