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Plotting a function graph/ x*exp(x)

Graphing y = x*exp(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          x
f(x) = x*e
f(x)=xexf{\left(x \right)} = x e^{x} 
f = x*exp(x)
The graph of the function
02468-8-6-4-2-1010-250000250000
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:
xex=0x e^{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−41.6261544568938x_{1} = -41.6261544568938
x2=−87.1541152286569x_{2} = -87.1541152286569
x3=−95.1266472537626x_{3} = -95.1266472537626
x4=−67.2586229734047x_{4} = -67.2586229734047
x5=−111.085180982879x_{5} = -111.085180982879
x6=−109.089608132217x_{6} = -109.089608132217
x7=−99.1148331129772x_{7} = -99.1148331129772
x8=−71.2319064024203x_{8} = -71.2319064024203
x9=−45.5287883412543x_{9} = -45.5287883412543
x10=−113.080930865701x_{10} = -113.080930865701
x11=−115.076847342498x_{11} = -115.076847342498
x12=−81.1789726997072x_{12} = -81.1789726997072
x13=−53.3950840173982x_{13} = -53.3950840173982
x14=−55.369883839131x_{14} = -55.369883839131
x15=−91.1396752246407x_{15} = -91.1396752246407
x16=−32.0913241206348x_{16} = -32.0913241206348
x17=−57.3470343910748x_{17} = -57.3470343910748
x18=−33.9540517145623x_{18} = -33.9540517145623
x19=−107.094223645316x_{19} = -107.094223645316
x20=−39.6870583075465x_{20} = -39.6870583075465
x21=−49.4541901054407x_{21} = -49.4541901054407
x22=−69.2447823410302x_{22} = -69.2447823410302
x23=−75.2086687051389x_{23} = -75.2086687051389
x24=−35.8463765939876x_{24} = -35.8463765939876
x25=−43.5740005056864x_{25} = -43.5740005056864
x26=−83.1702113647074x_{26} = -83.1702113647074
x27=−89.146704685936x_{27} = -89.146704685936
x28=−119.06914228288x_{28} = -119.06914228288
x29=−97.1205993527235x_{29} = -97.1205993527235
x30=−47.4891864944529x_{30} = -47.4891864944529
x31=−61.3071694941258x_{31} = -61.3071694941258
x32=−63.2896724119287x_{32} = -63.2896724119287
x33=−121.065503606275x_{33} = -121.065503606275
x34=−73.2198969347223x_{34} = -73.2198969347223
x35=−103.10407015753x_{35} = -103.10407015753
x36=−93.1329980618501x_{36} = -93.1329980618501
x37=−85.1619388762717x_{37} = -85.1619388762717
x38=−117.072920781941x_{38} = -117.072920781941
x39=−77.1981473783759x_{39} = -77.1981473783759
x40=−65.2735421114241x_{40} = -65.2735421114241
x41=−101.109329237227x_{41} = -101.109329237227
x42=−79.1882678183563x_{42} = -79.1882678183563
x43=−105.099039845199x_{43} = -105.099039845199
x44=0x_{44} = 0
x45=−59.3262172000187x_{45} = -59.3262172000187
x46=−51.4230249783974x_{46} = -51.4230249783974
x47=−37.7592416454249x_{47} = -37.7592416454249
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*exp(x).
0e00 e^{0}
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
xex+ex=0x e^{x} + e^{x} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1
The values of the extrema at the points:
       -1 
(-1, -e  )


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=−1x_{1} = -1
The function has no maxima
Decreasing at intervals
[−1,∞)\left[-1, \infty\right)
Increasing at intervals
(−∞,−1]\left(-\infty, -1\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
(x+2)ex=0\left(x + 2\right) e^{x} = 0
Solve this equation
The roots of this equation
x1=−2x_{1} = -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
[−2,∞)\left[-2, \infty\right)
Convex at the intervals
(−∞,−2]\left(-\infty, -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→−∞(xex)=0\lim_{x \to -\infty}\left(x e^{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(xex)=∞\lim_{x \to \infty}\left(x e^{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(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞ex=0\lim_{x \to -\infty} e^{x} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞ex=∞\lim_{x \to \infty} e^{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:
xex=−xe−xx e^{x} = - x e^{- x}
- No
xex=xe−xx e^{x} = x e^{- x}
- No
so, the function
not is
neither even, nor odd
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