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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
0-90-80-70-60-50-40-30-20-10-1000.00.5
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=0- x e^{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−53.3950840173982x_{1} = -53.3950840173982
x2=−65.2735421114241x_{2} = -65.2735421114241
x3=−47.4891864944529x_{3} = -47.4891864944529
x4=−73.2198969347223x_{4} = -73.2198969347223
x5=−93.1329980618501x_{5} = -93.1329980618501
x6=−75.2086687051389x_{6} = -75.2086687051389
x7=−121.065503606275x_{7} = -121.065503606275
x8=−39.6870583075465x_{8} = -39.6870583075465
x9=−45.5287883412543x_{9} = -45.5287883412543
x10=−87.1541152286569x_{10} = -87.1541152286569
x11=−71.2319064024203x_{11} = -71.2319064024203
x12=−83.1702113647074x_{12} = -83.1702113647074
x13=−55.369883839131x_{13} = -55.369883839131
x14=−109.089608132217x_{14} = -109.089608132217
x15=−43.5740005056864x_{15} = -43.5740005056864
x16=−115.076847342498x_{16} = -115.076847342498
x17=−119.06914228288x_{17} = -119.06914228288
x18=−103.10407015753x_{18} = -103.10407015753
x19=−91.1396752246407x_{19} = -91.1396752246407
x20=−117.072920781941x_{20} = -117.072920781941
x21=−33.9540517145623x_{21} = -33.9540517145623
x22=−113.080930865701x_{22} = -113.080930865701
x23=−111.085180982879x_{23} = -111.085180982879
x24=−49.4541901054407x_{24} = -49.4541901054407
x25=−61.3071694941258x_{25} = -61.3071694941258
x26=0x_{26} = 0
x27=−89.146704685936x_{27} = -89.146704685936
x28=−59.3262172000187x_{28} = -59.3262172000187
x29=−67.2586229734047x_{29} = -67.2586229734047
x30=−51.4230249783974x_{30} = -51.4230249783974
x31=−95.1266472537626x_{31} = -95.1266472537626
x32=−37.7592416454249x_{32} = -37.7592416454249
x33=−85.1619388762717x_{33} = -85.1619388762717
x34=−105.099039845199x_{34} = -105.099039845199
x35=−69.2447823410302x_{35} = -69.2447823410302
x36=−101.109329237227x_{36} = -101.109329237227
x37=−99.1148331129772x_{37} = -99.1148331129772
x38=−79.1882678183563x_{38} = -79.1882678183563
x39=−41.6261544568938x_{39} = -41.6261544568938
x40=−35.8463765939876x_{40} = -35.8463765939876
x41=−32.0913241206348x_{41} = -32.0913241206348
x42=−77.1981473783759x_{42} = -77.1981473783759
x43=−97.1205993527235x_{43} = -97.1205993527235
x44=−107.094223645316x_{44} = -107.094223645316
x45=−81.1789726997072x_{45} = -81.1789726997072
x46=−63.2896724119287x_{46} = -63.2896724119287
x47=−57.3470343910748x_{47} = -57.3470343910748
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).
−0e0- 0 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=0- x 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:
The function has no minima
Maxima of the function at points:
x1=−1x_{1} = -1
Decreasing at intervals
(−∞,−1]\left(-\infty, -1\right]
Increasing at intervals
[−1,∞)\left[-1, \infty\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(-\infty, -2\right]
Convex at the intervals
[−2,∞)\left[-2, \infty\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}\left(- e^{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−ex)=−∞\lim_{x \to \infty}\left(- e^{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:
−xex=xe−x- x e^{x} = x e^{- x}
- No
−xex=−xe−x- x e^{x} = - x e^{- x}
- No
so, the function
not is
neither even, nor odd
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