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  • How to use it?

  • Graphing y =:
  • x^4-4x^3+4
  • x^4-2x^3+4
  • x^4-2x^3+3
  • x^4-2

  • Identical expressions

  • x^ five *exp(x)
  • x to the power of 5 multiply by exponent of (x)
  • x to the power of five multiply by exponent of (x)
  • x5*exp(x)
  • x5*expx
  • x⁵*exp(x)
  • x^5exp(x)
  • x5exp(x)
  • x5expx
  • x^5expx
Plotting a function graph/ x^5*exp(x)

Graphing y = x^5*exp(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        5  x
f(x) = x *e
f(x)=x5exf{\left(x \right)} = x^{5} e^{x} 
f = x^5*exp(x)
The graph of the function
02468-8-6-4-2-1010-25000000002500000000
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:
x5ex=0x^{5} e^{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−55.7053138027862x_{1} = -55.7053138027862
x2=−59.4301507848393x_{2} = -59.4301507848393
x3=−63.203700759395x_{3} = -63.203700759395
x4=−52.0468879719824x_{4} = -52.0468879719824
x5=−68.9303498571571x_{5} = -68.9303498571571
x6=0x_{6} = 0
x7=−70.852928212213x_{7} = -70.852928212213
x8=−53.8663749209093x_{8} = -53.8663749209093
x9=−102.10838811719x_{9} = -102.10838811719
x10=−113.955573445251x_{10} = -113.955573445251
x11=−78.5938206116598x_{11} = -78.5938206116598
x12=−108.026952287924x_{12} = -108.026952287924
x13=−92.272998566361x_{13} = -92.272998566361
x14=−119.89249785946x_{14} = -119.89249785946
x15=−96.2021656600697x_{15} = -96.2021656600697
x16=−76.6520353100444x_{16} = -76.6520353100444
x17=−115.933700055035x_{17} = -115.933700055035
x18=−57.5607081773421x_{18} = -57.5607081773421
x19=−86.3946014159142x_{19} = -86.3946014159142
x20=−48.4824588083384x_{20} = -48.4824588083384
x21=−65.10486557711x_{21} = -65.10486557711
x22=−74.714319096109x_{22} = -74.714319096109
x23=−121.873072311935x_{23} = -121.873072311935
x24=−94.2366645482349x_{24} = -94.2366645482349
x25=−117.912691620425x_{25} = -117.912691620425
x26=−61.3116844143123x_{26} = -61.3116844143123
x27=−72.78111392386x_{27} = -72.78111392386
x28=−104.079997294078x_{28} = -104.079997294078
x29=−46.7486890118024x_{29} = -46.7486890118024
x30=−82.4881017429899x_{30} = -82.4881017429899
x31=−106.052879934533x_{31} = -106.052879934533
x32=−88.3517901497362x_{32} = -88.3517901497362
x33=−98.1693663786158x_{33} = -98.1693663786158
x34=−67.0140624380037x_{34} = -67.0140624380037
x35=−100.138144200895x_{35} = -100.138144200895
x36=−90.3113180352068x_{36} = -90.3113180352068
x37=−80.5392889805181x_{37} = -80.5392889805181
x38=−111.978366295904x_{38} = -111.978366295904
x39=−84.4399605227193x_{39} = -84.4399605227193
x40=−50.2506338869203x_{40} = -50.2506338869203
x41=−110.002137787397x_{41} = -110.002137787397
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^5*exp(x).
05e00^{5} 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
x5ex+5x4ex=0x^{5} e^{x} + 5 x^{4} e^{x} = 0
Solve this equation
The roots of this equation
x1=−5x_{1} = -5
x2=0x_{2} = 0
The values of the extrema at the points:
            -5 
(-5, -3125*e  )

(0, 0)


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=−5x_{1} = -5
The function has no maxima
Decreasing at intervals
[−5,∞)\left[-5, \infty\right)
Increasing at intervals
(−∞,−5]\left(-\infty, -5\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
x3(x2+10x+20)ex=0x^{3} \left(x^{2} + 10 x + 20\right) e^{x} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−5−5x_{2} = -5 - \sqrt{5}
x3=−5+5x_{3} = -5 + \sqrt{5}

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
[−5−5,−5+5]∪[0,∞)\left[-5 - \sqrt{5}, -5 + \sqrt{5}\right] \cup \left[0, \infty\right)
Convex at the intervals
(−∞,−5−5]∪[−5+5,0]\left(-\infty, -5 - \sqrt{5}\right] \cup \left[-5 + \sqrt{5}, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x5ex)=0\lim_{x \to -\infty}\left(x^{5} e^{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(x5ex)=∞\lim_{x \to \infty}\left(x^{5} 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^5*exp(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x4ex)=0\lim_{x \to -\infty}\left(x^{4} e^{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(x4ex)=∞\lim_{x \to \infty}\left(x^{4} 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:
x5ex=−x5e−xx^{5} e^{x} = - x^{5} e^{- x}
- No
x5ex=x5e−xx^{5} e^{x} = x^{5} e^{- x}
- No
so, the function
not is
neither even, nor odd
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