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Plotting a function graph/ x*e^(3x)

Graphing y = x*e^(3x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          3*x
f(x) = x*E
f(x)=e3xxf{\left(x \right)} = e^{3 x} x 
f = E^(3*x)*x
The graph of the function
02468-8-6-4-2-1010200000000000000-100000000000000
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:
e3xx=0e^{3 x} x = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−32.9486642444431x_{1} = -32.9486642444431
x2=−74.9005485194854x_{2} = -74.9005485194854
x3=−86.895779213308x_{3} = -86.895779213308
x4=−36.9387450640686x_{4} = -36.9387450640686
x5=−46.9220965396229x_{5} = -46.9220965396229
x6=−38.9346459151651x_{6} = -38.9346459151651
x7=−92.8938781666278x_{7} = -92.8938781666278
x8=−94.8933003357084x_{8} = -94.8933003357084
x9=−70.9025193770283x_{9} = -70.9025193770283
x10=0x_{10} = 0
x11=−13.1416111389768x_{11} = -13.1416111389768
x12=−24.98010213912x_{12} = -24.98010213912
x13=−21.0073010205309x_{13} = -21.0073010205309
x14=−66.9047415605498x_{14} = -66.9047415605498
x15=−17.0522033058526x_{15} = -17.0522033058526
x16=−34.9433802056x_{16} = -34.9433802056
x17=−48.9196619333906x_{17} = -48.9196619333906
x18=−19.0265962551348x_{18} = -19.0265962551348
x19=−28.9618160346074x_{19} = -28.9618160346074
x20=−106.890305626567x_{20} = -106.890305626567
x21=−98.8922178388967x_{21} = -98.8922178388967
x22=−60.9086624652399x_{22} = -60.9086624652399
x23=−88.8951156279565x_{23} = -88.8951156279565
x24=−96.8927474190871x_{24} = -96.8927474190871
x25=−68.9035961391333x_{25} = -68.9035961391333
x26=−30.9547443398454x_{26} = -30.9547443398454
x27=−42.9277215719632x_{27} = -42.9277215719632
x28=−26.9701450522659x_{28} = -26.9701450522659
x29=−76.8996443728558x_{29} = -76.8996443728558
x30=−100.891710147796x_{30} = -100.891710147796
x31=−62.9072664632937x_{31} = -62.9072664632937
x32=−15.0879331724686x_{32} = -15.0879331724686
x33=−54.91351121676x_{33} = -54.91351121676
x34=−52.9153929935434x_{34} = -52.9153929935434
x35=−72.9015052766187x_{35} = -72.9015052766187
x36=−40.9309946683666x_{36} = -40.9309946683666
x37=−50.9174360047675x_{37} = -50.9174360047675
x38=−78.8987886111081x_{38} = -78.8987886111081
x39=−56.9117722918857x_{39} = -56.9117722918857
x40=−84.8964756651569x_{40} = -84.8964756651569
x41=−64.9059624301828x_{41} = -64.9059624301828
x42=−11.2328152835375x_{42} = -11.2328152835375
x43=−80.8979774495789x_{43} = -80.8979774495789
x44=−90.894482635299x_{44} = -90.894482635299
x45=−102.891223015744x_{45} = -102.891223015744
x46=−104.890755218345x_{46} = -104.890755218345
x47=−22.9922211573021x_{47} = -22.9922211573021
x48=−58.9101605312548x_{48} = -58.9101605312548
x49=−82.8972074884783x_{49} = -82.8972074884783
x50=−44.9247706684296x_{50} = -44.9247706684296
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*E^(3*x).
0e0⋅30 e^{0 \cdot 3}
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
3xe3x+e3x=03 x e^{3 x} + e^{3 x} = 0
Solve this equation
The roots of this equation
x1=−13x_{1} = - \frac{1}{3}
The values of the extrema at the points:
         -1  
       -e    
(-1/3, -----)
         3


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=−13x_{1} = - \frac{1}{3}
The function has no maxima
Decreasing at intervals
[−13,∞)\left[- \frac{1}{3}, \infty\right)
Increasing at intervals
(−∞,−13]\left(-\infty, - \frac{1}{3}\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
3(3x+2)e3x=03 \left(3 x + 2\right) e^{3 x} = 0
Solve this equation
The roots of this equation
x1=−23x_{1} = - \frac{2}{3}

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
[−23,∞)\left[- \frac{2}{3}, \infty\right)
Convex at the intervals
(−∞,−23]\left(-\infty, - \frac{2}{3}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(e3xx)=0\lim_{x \to -\infty}\left(e^{3 x} x\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(e3xx)=∞\lim_{x \to \infty}\left(e^{3 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^(3*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞e3x=0\lim_{x \to -\infty} e^{3 x} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞e3x=∞\lim_{x \to \infty} e^{3 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:
e3xx=−xe−3xe^{3 x} x = - x e^{- 3 x}
- No
e3xx=xe−3xe^{3 x} x = x e^{- 3 x}
- No
so, the function
not is
neither even, nor odd
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