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Plotting a function graph/ xe^(6x)

Graphing y = xe^(6x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          6*x
f(x) = x*E
f(x)=e6xxf{\left(x \right)} = e^{6 x} x 
f = E^(6*x)*x
The graph of the function
02468-8-6-4-2-10102e27-1e27
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:
e6xx=0e^{6 x} x = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−89.3862335187239x_{1} = -89.3862335187239
x2=−49.3918270477381x_{2} = -49.3918270477381
x3=−71.387952162544x_{3} = -71.387952162544
x4=−5.56389762880346x_{4} = -5.56389762880346
x5=−53.3908755563758x_{5} = -53.3908755563758
x6=−21.4095189464519x_{6} = -21.4095189464519
x7=−41.3943104955653x_{7} = -41.3943104955653
x8=−31.3993199729743x_{8} = -31.3993199729743
x9=−15.4230177001815x_{9} = -15.4230177001815
x10=−29.4007619756022x_{10} = -29.4007619756022
x11=−99.3855533022583x_{11} = -99.3855533022583
x12=0x_{12} = 0
x13=−101.385433688296x_{13} = -101.385433688296
x14=−55.3904533198466x_{14} = -55.3904533198466
x15=−85.3865512986458x_{15} = -85.3865512986458
x16=−13.4307238408575x_{16} = -13.4307238408575
x17=−45.3929541627989x_{17} = -45.3929541627989
x18=−57.3900615984483x_{18} = -57.3900615984483
x19=−7.4899952959196x_{19} = -7.4899952959196
x20=−61.3893573582287x_{20} = -61.3893573582287
x21=−69.3881998218577x_{21} = -69.3881998218577
x22=−79.3870895834467x_{22} = -79.3870895834467
x23=−35.3969553852839x_{23} = -35.3969553852839
x24=−47.3923657462765x_{24} = -47.3923657462765
x25=−103.385318794352x_{25} = -103.385318794352
x26=−19.4129751413295x_{26} = -19.4129751413295
x27=−37.3959739307627x_{27} = -37.3959739307627
x28=−65.3887420484348x_{28} = -65.3887420484348
x29=−67.388462638586x_{29} = -67.388462638586
x30=−25.4043884996046x_{30} = -25.4043884996046
x31=−83.3867219056926x_{31} = -83.3867219056926
x32=−81.3869011116302x_{32} = -81.3869011116302
x33=−95.385807903624x_{33} = -95.385807903624
x34=−87.3863886860237x_{34} = -87.3863886860237
x35=−75.3874973564366x_{35} = -75.3874973564366
x36=−59.389697198815x_{36} = -59.389697198815
x37=−17.4173374335761x_{37} = -17.4173374335761
x38=−39.3950976523916x_{38} = -39.3950976523916
x39=−97.3856779332234x_{39} = -97.3856779332234
x40=−91.3860852968104x_{40} = -91.3860852968104
x41=−63.3890396743785x_{41} = -63.3890396743785
x42=−23.4067126659642x_{42} = -23.4067126659642
x43=−73.3877183859508x_{43} = -73.3877183859508
x44=−43.3935995163912x_{44} = -43.3935995163912
x45=−9.45905405999644x_{45} = -9.45905405999644
x46=−27.4024318989858x_{46} = -27.4024318989858
x47=−51.3913320193907x_{47} = -51.3913320193907
x48=−11.4417869588292x_{48} = -11.4417869588292
x49=−33.398062176249x_{49} = -33.398062176249
x50=−77.3872880589524x_{50} = -77.3872880589524
x51=−93.3859435641304x_{51} = -93.3859435641304
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*E^(6*x).
0e0⋅60 e^{0 \cdot 6}
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
6xe6x+e6x=06 x e^{6 x} + e^{6 x} = 0
Solve this equation
The roots of this equation
x1=−16x_{1} = - \frac{1}{6}
The values of the extrema at the points:
         -1  
       -e    
(-1/6, -----)
         6


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=−16x_{1} = - \frac{1}{6}
The function has no maxima
Decreasing at intervals
[−16,∞)\left[- \frac{1}{6}, \infty\right)
Increasing at intervals
(−∞,−16]\left(-\infty, - \frac{1}{6}\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
12(3x+1)e6x=012 \left(3 x + 1\right) e^{6 x} = 0
Solve this equation
The roots of this equation
x1=−13x_{1} = - \frac{1}{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
[−13,∞)\left[- \frac{1}{3}, \infty\right)
Convex at the intervals
(−∞,−13]\left(-\infty, - \frac{1}{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→−∞(e6xx)=0\lim_{x \to -\infty}\left(e^{6 x} x\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(e6xx)=∞\lim_{x \to \infty}\left(e^{6 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^(6*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞e6x=0\lim_{x \to -\infty} e^{6 x} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞e6x=∞\lim_{x \to \infty} e^{6 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:
e6xx=−xe−6xe^{6 x} x = - x e^{- 6 x}
- No
e6xx=xe−6xe^{6 x} x = x e^{- 6 x}
- No
so, the function
not is
neither even, nor odd
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