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  • How to use it?

  • Graphing y =:
  • x^3-x^2-2x
  • |x+3|-|x-1|+x+2
  • (x-3)sqrt(x)
  • |x|-3
  • Integral of d{x}:
  • x^2*exp(2*x) x^2*exp(2*x)

  • Identical expressions

  • x^ two *exp(two *x)
  • x squared multiply by exponent of (2 multiply by x)
  • x to the power of two multiply by exponent of (two multiply by x)
  • x2*exp(2*x)
  • x2*exp2*x
  • x²*exp(2*x)
  • x to the power of 2*exp(2*x)
  • x^2exp(2x)
  • x2exp(2x)
  • x2exp2x
  • x^2exp2x
Plotting a function graph/ x^2*exp(2*x)

Graphing y = x^2*exp(2*x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        2  2*x
f(x) = x *e
f(x)=x2e2xf{\left(x \right)} = x^{2} e^{2 x} 
f = x^2*exp(2*x)
The graph of the function
02468-8-6-4-2-1010050000000000
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:
x2e2x=0x^{2} e^{2 x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−36.7195320816298x_{1} = -36.7195320816298
x2=−26.9010959349039x_{2} = -26.9010959349039
x3=−82.504650810262x_{3} = -82.504650810262
x4=−56.5770897316422x_{4} = -56.5770897316422
x5=−21.1434296492342x_{5} = -21.1434296492342
x6=−50.6060724942638x_{6} = -50.6060724942638
x7=−76.5165673980131x_{7} = -76.5165673980131
x8=−48.6175931384471x_{8} = -48.6175931384471
x9=−104.473479319738x_{9} = -104.473479319738
x10=−54.5859383557408x_{10} = -54.5859383557408
x11=−28.8510959907969x_{11} = -28.8510959907969
x12=−108.469260442292x_{12} = -108.469260442292
x13=−80.5084073672939x_{13} = -80.5084073672939
x14=−68.5360631673775x_{14} = -68.5360631673775
x15=−70.5307298408637x_{15} = -70.5307298408637
x16=−30.8097250983357x_{16} = -30.8097250983357
x17=−17.4921761992353x_{17} = -17.4921761992353
x18=−66.5417541529036x_{18} = -66.5417541529036
x19=−19.2843038712141x_{19} = -19.2843038712141
x20=−100.478058284678x_{20} = -100.478058284678
x21=−46.6302794426095x_{21} = -46.6302794426095
x22=−32.7749096508693x_{22} = -32.7749096508693
x23=−40.6774272261348x_{23} = -40.6774272261348
x24=−98.4804975285776x_{24} = -98.4804975285776
x25=−62.5543635916052x_{25} = -62.5543635916052
x26=−88.4944847627286x_{26} = -88.4944847627286
x27=−90.4914199700898x_{27} = -90.4914199700898
x28=−64.5478400761828x_{28} = -64.5478400761828
x29=−52.5955635451267x_{29} = -52.5955635451267
x30=−72.5257214483396x_{30} = -72.5257214483396
x31=−42.6599394398514x_{31} = -42.6599394398514
x32=−34.7451956711642x_{32} = -34.7451956711642
x33=−110.467273013183x_{33} = -110.467273013183
x34=−78.5123735412569x_{34} = -78.5123735412569
x35=−38.6971394656718x_{35} = -38.6971394656718
x36=−96.4830455203986x_{36} = -96.4830455203986
x37=−94.4857097027622x_{37} = -94.4857097027622
x38=−84.501087667802x_{38} = -84.501087667802
x39=−58.5689271719993x_{39} = -58.5689271719993
x40=−86.4977033666426x_{40} = -86.4977033666426
x41=0x_{41} = 0
x42=−23.0409545610204x_{42} = -23.0409545610204
x43=−106.471327579279x_{43} = -106.471327579279
x44=−24.962794643797x_{44} = -24.962794643797
x45=−60.5613737361617x_{45} = -60.5613737361617
x46=−102.475720968758x_{46} = -102.475720968758
x47=−92.4884982137411x_{47} = -92.4884982137411
x48=−74.5210091430628x_{48} = -74.5210091430628
x49=−44.6443184018587x_{49} = -44.6443184018587
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^2*exp(2*x).
02e0⋅20^{2} 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
2x2e2x+2xe2x=02 x^{2} e^{2 x} + 2 x e^{2 x} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1
x2=0x_{2} = 0
The values of the extrema at the points:
      -2 
(-1, 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=0x_{1} = 0
Maxima of the function at points:
x1=−1x_{1} = -1
Decreasing at intervals
(−∞,−1]∪[0,∞)\left(-\infty, -1\right] \cup \left[0, \infty\right)
Increasing at intervals
[−1,0]\left[-1, 0\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
2(2x2+4x+1)e2x=02 \left(2 x^{2} + 4 x + 1\right) e^{2 x} = 0
Solve this equation
The roots of this equation
x1=−1−22x_{1} = -1 - \frac{\sqrt{2}}{2}
x2=−1+22x_{2} = -1 + \frac{\sqrt{2}}{2}

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
(−∞,−1−22]∪[−1+22,∞)\left(-\infty, -1 - \frac{\sqrt{2}}{2}\right] \cup \left[-1 + \frac{\sqrt{2}}{2}, \infty\right)
Convex at the intervals
[−1−22,−1+22]\left[-1 - \frac{\sqrt{2}}{2}, -1 + \frac{\sqrt{2}}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x2e2x)=0\lim_{x \to -\infty}\left(x^{2} e^{2 x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(x2e2x)=∞\lim_{x \to \infty}\left(x^{2} 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^2*exp(2*x), divided by x 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,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(xe2x)=∞\lim_{x \to \infty}\left(x e^{2 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:
x2e2x=x2e−2xx^{2} e^{2 x} = x^{2} e^{- 2 x}
- No
x2e2x=−x2e−2xx^{2} e^{2 x} = - x^{2} e^{- 2 x}
- No
so, the function
not is
neither even, nor odd
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