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Plotting a function graph/ x*e^x

Graphing y = x*e^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)=exxf{\left(x \right)} = e^{x} x 
f = E^x*x
The graph of the function
0-80-60-40-2020406080-1001005e45-3e45
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:
exx=0e^{x} x = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−45.5287883412543x_{1} = -45.5287883412543
x2=−37.7592416454249x_{2} = -37.7592416454249
x3=−55.369883839131x_{3} = -55.369883839131
x4=−103.10407015753x_{4} = -103.10407015753
x5=−95.1266472537626x_{5} = -95.1266472537626
x6=−115.076847342498x_{6} = -115.076847342498
x7=0x_{7} = 0
x8=−71.2319064024203x_{8} = -71.2319064024203
x9=−83.1702113647074x_{9} = -83.1702113647074
x10=−119.06914228288x_{10} = -119.06914228288
x11=−49.4541901054407x_{11} = -49.4541901054407
x12=−51.4230249783974x_{12} = -51.4230249783974
x13=−117.072920781941x_{13} = -117.072920781941
x14=−39.6870583075465x_{14} = -39.6870583075465
x15=−99.1148331129772x_{15} = -99.1148331129772
x16=−91.1396752246407x_{16} = -91.1396752246407
x17=−97.1205993527235x_{17} = -97.1205993527235
x18=−73.2198969347223x_{18} = -73.2198969347223
x19=−32.0913241206348x_{19} = -32.0913241206348
x20=−65.2735421114241x_{20} = -65.2735421114241
x21=−105.099039845199x_{21} = -105.099039845199
x22=−79.1882678183563x_{22} = -79.1882678183563
x23=−63.2896724119287x_{23} = -63.2896724119287
x24=−57.3470343910748x_{24} = -57.3470343910748
x25=−75.2086687051389x_{25} = -75.2086687051389
x26=−47.4891864944529x_{26} = -47.4891864944529
x27=−111.085180982879x_{27} = -111.085180982879
x28=−121.065503606275x_{28} = -121.065503606275
x29=−67.2586229734047x_{29} = -67.2586229734047
x30=−113.080930865701x_{30} = -113.080930865701
x31=−61.3071694941258x_{31} = -61.3071694941258
x32=−53.3950840173982x_{32} = -53.3950840173982
x33=−77.1981473783759x_{33} = -77.1981473783759
x34=−59.3262172000187x_{34} = -59.3262172000187
x35=−109.089608132217x_{35} = -109.089608132217
x36=−81.1789726997072x_{36} = -81.1789726997072
x37=−35.8463765939876x_{37} = -35.8463765939876
x38=−41.6261544568938x_{38} = -41.6261544568938
x39=−89.146704685936x_{39} = -89.146704685936
x40=−101.109329237227x_{40} = -101.109329237227
x41=−43.5740005056864x_{41} = -43.5740005056864
x42=−87.1541152286569x_{42} = -87.1541152286569
x43=−107.094223645316x_{43} = -107.094223645316
x44=−69.2447823410302x_{44} = -69.2447823410302
x45=−85.1619388762717x_{45} = -85.1619388762717
x46=−93.1329980618501x_{46} = -93.1329980618501
x47=−33.9540517145623x_{47} = -33.9540517145623
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*E^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
ex+xex=0e^{x} + 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→−∞(exx)=0\lim_{x \to -\infty}\left(e^{x} x\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(exx)=∞\lim_{x \to \infty}\left(e^{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^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:
exx=−xe−xe^{x} x = - x e^{- x}
- No
exx=xe−xe^{x} x = x e^{- x}
- No
so, the function
not is
neither even, nor odd
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