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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)=e−xxf{\left(x \right)} = e^{- x} x 
f = E^(-x)*x
The graph of the function
01020304050607080901000.00.5
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:
e−xx=0e^{- x} x = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=115.385891060967x_{1} = 115.385891060967
x2=87.4626045093137x_{2} = 87.4626045093137
x3=55.67586733869x_{3} = 55.67586733869
x4=89.4552548670559x_{4} = 89.4552548670559
x5=32.3772961851972x_{5} = 32.3772961851972
x6=79.496455118891x_{6} = 79.496455118891
x7=107.40315817241x_{7} = 107.40315817241
x8=59.6328238138969x_{8} = 59.6328238138969
x9=117.381987933686x_{9} = 117.381987933686
x10=73.5277731870455x_{10} = 73.5277731870455
x11=41.9272307499711x_{11} = 41.9272307499711
x12=111.394173451874x_{12} = 111.394173451874
x13=69.5523925194344x_{13} = 69.5523925194344
x14=39.9866376954424x_{14} = 39.9866376954424
x15=97.429350983852x_{15} = 97.429350983852
x16=75.5166588459953x_{16} = 75.5166588459953
x17=71.5396566043977x_{17} = 71.5396566043977
x18=61.614029218278x_{18} = 61.614029218278
x19=105.407942520376x_{19} = 105.407942520376
x20=113.389949729147x_{20} = 113.389949729147
x21=93.4416565533312x_{21} = 93.4416565533312
x22=0x_{22} = 0
x23=81.4872456640903x_{23} = 81.4872456640903
x24=36.1413894508705x_{24} = 36.1413894508705
x25=65.580821222158x_{25} = 65.580821222158
x26=51.7281686335153x_{26} = 51.7281686335153
x27=38.0568716419232x_{27} = 38.0568716419232
x28=121.374613775997x_{28} = 121.374613775997
x29=49.758798960419x_{29} = 49.758798960419
x30=67.5660769899711x_{30} = 67.5660769899711
x31=43.8762545098096x_{31} = 43.8762545098096
x32=91.4482816547886x_{32} = 91.4482816547886
x33=85.4703620749206x_{33} = 85.4703620749206
x34=34.2454094695441x_{34} = 34.2454094695441
x35=83.4785626915261x_{35} = 83.4785626915261
x36=45.8319875396224x_{36} = 45.8319875396224
x37=101.418161552262x_{37} = 101.418161552262
x38=63.5967547129854x_{38} = 63.5967547129854
x39=119.378231552779x_{39} = 119.378231552779
x40=47.7931569932505x_{40} = 47.7931569932505
x41=109.398572537176x_{41} = 109.398572537176
x42=57.6533514231885x_{42} = 57.6533514231885
x43=77.5062407712727x_{43} = 77.5062407712727
x44=95.4353540260187x_{44} = 95.4353540260187
x45=53.7006804984823x_{45} = 53.7006804984823
x46=99.4236264980399x_{46} = 99.4236264980399
x47=103.412938828373x_{47} = 103.412938828373
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).
0e−00 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
−xe−x+e−x=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)e−x=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→−∞(e−xx)=−∞\lim_{x \to -\infty}\left(e^{- x} x\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(e−xx)=0\lim_{x \to \infty}\left(e^{- x} x\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
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→−∞e−x=∞\lim_{x \to -\infty} e^{- x} = \infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞e−x=0\lim_{x \to \infty} e^{- x} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
e−xx=−xexe^{- x} x = - x e^{x}
- No
e−xx=xexe^{- x} x = x e^{x}
- No
so, the function
not is
neither even, nor odd
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