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Plotting a function graph/ (1+x*(1+x))*exp(-x)

Graphing y = (1+x*(1+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) = (1 + x*(1 + x))*e
f(x)=(x(x+1)+1)e−xf{\left(x \right)} = \left(x \left(x + 1\right) + 1\right) e^{- x} 
f = (x*(x + 1) + 1)*exp(-x)
The graph of the function
02468-8-6-4-2-101002000000
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:
(x(x+1)+1)e−x=0\left(x \left(x + 1\right) + 1\right) e^{- x} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=97.6792665182734x_{1} = 97.6792665182734
x2=101.656718461289x_{2} = 101.656718461289
x3=87.7462973895715x_{3} = 87.7462973895715
x4=75.286424855108x_{4} = 75.286424855108
x5=60.0896238087298x_{5} = 60.0896238087298
x6=107.626491356935x_{6} = 107.626491356935
x7=73.8777249104058x_{7} = 73.8777249104058
x8=99.6677304943578x_{8} = 99.6677304943578
x9=56.1764128472192x_{9} = 56.1764128472192
x10=85.7619385838247x_{10} = 85.7619385838247
x11=48.4126335633387x_{11} = 48.4126335633387
x12=50.3434895067961x_{12} = 50.3434895067961
x13=77.8342933423609x_{13} = 77.8342933423609
x14=71.9016957478322x_{14} = 71.9016957478322
x15=81.7959842371517x_{15} = 81.7959842371517
x16=115.591713770949x_{16} = 115.591713770949
x17=95.6913649521561x_{17} = 95.6913649521561
x18=93.7040680768252x_{18} = 93.7040680768252
x19=111.608393846173x_{19} = 111.608393846173
x20=79.8145571065428x_{20} = 79.8145571065428
x21=105.63612946398x_{21} = 105.63612946398
x22=69.9273869463763x_{22} = 69.9273869463763
x23=42.6818562990598x_{23} = 42.6818562990598
x24=119.576290618133x_{24} = 119.576290618133
x25=35.3149822710978x_{25} = 35.3149822710978
x26=58.1310183346547x_{26} = 58.1310183346547
x27=83.7784744169568x_{27} = 83.7784744169568
x28=37.1091027064473x_{28} = 37.1091027064473
x29=62.0517182834213x_{29} = 62.0517182834213
x30=52.2818070493597x_{30} = 52.2818070493597
x31=64.0168749663061x_{31} = 64.0168749663061
x32=65.9847344724224x_{32} = 65.9847344724224
x33=46.4907116808764x_{33} = 46.4907116808764
x34=54.2264246629927x_{34} = 54.2264246629927
x35=103.646195429544x_{35} = 103.646195429544
x36=44.5796215483367x_{36} = 44.5796215483367
x37=67.9549919282347x_{37} = 67.9549919282347
x38=117.583854144694x_{38} = 117.583854144694
x39=121.569006742071x_{39} = 121.569006742071
x40=40.8007664623713x_{40} = 40.8007664623713
x41=109.617254342522x_{41} = 109.617254342522
x42=75.8553063627584x_{42} = 75.8553063627584
x43=89.7314799061846x_{43} = 89.7314799061846
x44=113.599887264453x_{44} = 113.599887264453
x45=38.9409792435025x_{45} = 38.9409792435025
x46=91.7174225341436x_{46} = 91.7174225341436
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (1 + x*(1 + x))*exp(-x).
e−0e^{- 0}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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
(2x+1)e−x−(x(x+1)+1)e−x=0\left(2 x + 1\right) e^{- x} - \left(x \left(x + 1\right) + 1\right) e^{- x} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=1x_{2} = 1
The values of the extrema at the points:
(0, 1)

       -1 
(1, 3*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=0x_{1} = 0
Maxima of the function at points:
x1=1x_{1} = 1
Decreasing at intervals
[0,1]\left[0, 1\right]
Increasing at intervals
(−∞,0]∪[1,∞)\left(-\infty, 0\right] \cup \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(x+1)−4x+1)e−x=0\left(x \left(x + 1\right) - 4 x + 1\right) e^{- x} = 0
Solve this equation
The roots of this equation
x1=32−52x_{1} = \frac{3}{2} - \frac{\sqrt{5}}{2}
x2=52+32x_{2} = \frac{\sqrt{5}}{2} + \frac{3}{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
(−∞,32−52]∪[52+32,∞)\left(-\infty, \frac{3}{2} - \frac{\sqrt{5}}{2}\right] \cup \left[\frac{\sqrt{5}}{2} + \frac{3}{2}, \infty\right)
Convex at the intervals
[32−52,52+32]\left[\frac{3}{2} - \frac{\sqrt{5}}{2}, \frac{\sqrt{5}}{2} + \frac{3}{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→−∞((x(x+1)+1)e−x)=∞\lim_{x \to -\infty}\left(\left(x \left(x + 1\right) + 1\right) e^{- x}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞((x(x+1)+1)e−x)=0\lim_{x \to \infty}\left(\left(x \left(x + 1\right) + 1\right) e^{- 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 (1 + x*(1 + x))*exp(-x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x(x+1)+1)e−xx)=−∞\lim_{x \to -\infty}\left(\frac{\left(x \left(x + 1\right) + 1\right) e^{- x}}{x}\right) = -\infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞((x(x+1)+1)e−xx)=0\lim_{x \to \infty}\left(\frac{\left(x \left(x + 1\right) + 1\right) e^{- x}}{x}\right) = 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:
(x(x+1)+1)e−x=(−x(1−x)+1)ex\left(x \left(x + 1\right) + 1\right) e^{- x} = \left(- x \left(1 - x\right) + 1\right) e^{x}
- No
(x(x+1)+1)e−x=−(−x(1−x)+1)ex\left(x \left(x + 1\right) + 1\right) e^{- x} = - \left(- x \left(1 - x\right) + 1\right) e^{x}
- No
so, the function
not is
neither even, nor odd
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