Mister Exam

Other calculators

Graphing y = x^6*e^x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src]
        6  x
f(x) = x *E 
f(x)=exx6f{\left(x \right)} = e^{x} x^{6}
f = E^x*x^6
The graph of the function
02468-8-6-4-2-1010025000000000
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:
exx6=0e^{x} x^{6} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=78.9671108208535x_{1} = -78.9671108208535
x2=58.162685619902x_{2} = -58.162685619902
x3=110.24134484425x_{3} = -110.24134484425
x4=0x_{4} = 0
x5=96.4861171978203x_{5} = -96.4861171978203
x6=71.2864585440044x_{6} = -71.2864585440044
x7=106.303377433517x_{7} = -106.303377433517
x8=56.3423143698061x_{8} = -56.3423143698061
x9=120.107453884621x_{9} = -120.107453884621
x10=100.40770624888x_{10} = -100.40770624888
x11=116.15774602451x_{11} = -116.15774602451
x12=82.8370640708798x_{12} = -82.8370640708798
x13=114.184456627505x_{13} = -114.184456627505
x14=77.0387854017141x_{14} = -77.0387854017141
x15=67.4854672597338x_{15} = -67.4854672597338
x16=108.27167551581x_{16} = -108.27167551581
x17=54.5425970346382x_{17} = -54.5425970346382
x18=60.0006716761699x_{18} = -60.0006716761699
x19=65.5977440883992x_{19} = -65.5977440883992
x20=75.1155185220878x_{20} = -75.1155185220878
x21=73.1978637909425x_{21} = -73.1978637909425
x22=92.5729433217803x_{22} = -92.5729433217803
x23=112.21229846825x_{23} = -112.21229846825
x24=118.132099183578x_{24} = -118.132099183578
x25=61.8538036962972x_{25} = -61.8538036962972
x26=51.0212656989008x_{26} = -51.0212656989008
x27=84.7778953156122x_{27} = -84.7778953156122
x28=94.5283960776019x_{28} = -94.5283960776019
x29=86.7221756752623x_{29} = -86.7221756752623
x30=80.9000115859487x_{30} = -80.9000115859487
x31=122.083752661696x_{31} = -122.083752661696
x32=104.336545566176x_{32} = -104.336545566176
x33=90.6199460984893x_{33} = -90.6199460984893
x34=63.7200546034686x_{34} = -63.7200546034686
x35=102.371283845722x_{35} = -102.371283845722
x36=88.6696127131053x_{36} = -88.6696127131053
x37=52.7673199737009x_{37} = -52.7673199737009
x38=98.4459380391843x_{38} = -98.4459380391843
x39=69.382040298942x_{39} = -69.382040298942
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^6*E^x.
06e00^{6} 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
x6ex+6x5ex=0x^{6} e^{x} + 6 x^{5} e^{x} = 0
Solve this equation
The roots of this equation
x1=6x_{1} = -6
x2=0x_{2} = 0
The values of the extrema at the points:
            -6 
(-6, 46656*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=0x_{1} = 0
Maxima of the function at points:
x1=6x_{1} = -6
Decreasing at intervals
(,6][0,)\left(-\infty, -6\right] \cup \left[0, \infty\right)
Increasing at intervals
[6,0]\left[-6, 0\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
x4(x2+12x+30)ex=0x^{4} \left(x^{2} + 12 x + 30\right) e^{x} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=66x_{2} = -6 - \sqrt{6}
x3=6+6x_{3} = -6 + \sqrt{6}

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
(,66][6+6,)\left(-\infty, -6 - \sqrt{6}\right] \cup \left[-6 + \sqrt{6}, \infty\right)
Convex at the intervals
[66,6+6]\left[-6 - \sqrt{6}, -6 + \sqrt{6}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
limx(exx6)=0\lim_{x \to -\infty}\left(e^{x} x^{6}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
limx(exx6)=\lim_{x \to \infty}\left(e^{x} x^{6}\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^6*E^x, divided by x at x->+oo and x ->-oo
limx(x5ex)=0\lim_{x \to -\infty}\left(x^{5} e^{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
limx(x5ex)=\lim_{x \to \infty}\left(x^{5} 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:
exx6=x6exe^{x} x^{6} = x^{6} e^{- x}
- No
exx6=x6exe^{x} x^{6} = - x^{6} e^{- x}
- No
so, the function
not is
neither even, nor odd