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Plotting a function graph/ x^(1/x)

Graphing y = x^(1/x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       x ___
f(x) = \/ x
f(x)=x1xf{\left(x \right)} = x^{\frac{1}{x}} 
f = x^(1/x)
The graph of the function
02468-8-6-4-2-101002
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
x1x=0x^{\frac{1}{x}} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^(1/x).
0100^{\frac{1}{0}}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
x1x(−log⁡(x)x2+1x2)=0x^{\frac{1}{x}} \left(- \frac{\log{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}\right) = 0
Solve this equation
The roots of this equation
x1=ex_{1} = e
The values of the extrema at the points:
     / -1\ 
     \e  / 
(E, 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=ex_{1} = e
Decreasing at intervals
(−∞,e]\left(-\infty, e\right]
Increasing at intervals
[e,∞)\left[e, \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
x1x(2log⁡(x)−3+(log⁡(x)−1)2x)x3=0\frac{x^{\frac{1}{x}} \left(2 \log{\left(x \right)} - 3 + \frac{\left(\log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=36956.6362249476x_{1} = 36956.6362249476
x2=35842.5039425585x_{2} = 35842.5039425585
x3=43618.932582736x_{3} = 43618.932582736
x4=48041.0543039226x_{4} = 48041.0543039226
x5=38069.6475342003x_{5} = 38069.6475342003
x6=24632.3562385126x_{6} = 24632.3562385126
x7=52449.4275796145x_{7} = 52449.4275796145
x8=51348.5500563983x_{8} = 51348.5500563983
x9=44725.8295428241x_{9} = 44725.8295428241
x10=26885.164415468x_{10} = 26885.164415468
x11=4.36777096705602x_{11} = 4.36777096705602
x12=45831.7998080343x_{12} = 45831.7998080343
x13=41402.2587671693x_{13} = 41402.2587671693
x14=33610.7485850034x_{14} = 33610.7485850034
x15=40292.4295040997x_{15} = 40292.4295040997
x16=31374.1183597028x_{16} = 31374.1183597028
x17=30253.89094583x_{17} = 30253.89094583
x18=29132.3439864666x_{18} = 29132.3439864666
x19=56845.3273080066x_{19} = 56845.3273080066
x20=46936.8671226986x_{20} = 46936.8671226986
x21=28009.4451941101x_{21} = 28009.4451941101
x22=39181.5686882772x_{22} = 39181.5686882772
x23=42511.0842227899x_{23} = 42511.0842227899
x24=55747.4592642109x_{24} = 55747.4592642109
x25=57942.4847067835x_{25} = 57942.4847067835
x26=34727.2189284023x_{26} = 34727.2189284023
x27=49144.3832768866x_{27} = 49144.3832768866
x28=32493.05963509x_{28} = 32493.05963509
x29=53549.5263991257x_{29} = 53549.5263991257
x30=25759.4750752963x_{30} = 25759.4750752963
x31=54648.8645194423x_{31} = 54648.8645194423
x32=50246.8751110307x_{32} = 50246.8751110307
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0

lim⁡x→0−(x1x(2log⁡(x)−3+(log⁡(x)−1)2x)x3)=∞\lim_{x \to 0^-}\left(\frac{x^{\frac{1}{x}} \left(2 \log{\left(x \right)} - 3 + \frac{\left(\log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}}\right) = \infty
lim⁡x→0+(x1x(2log⁡(x)−3+(log⁡(x)−1)2x)x3)=0\lim_{x \to 0^+}\left(\frac{x^{\frac{1}{x}} \left(2 \log{\left(x \right)} - 3 + \frac{\left(\log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}}\right) = 0
- the limits are not equal, so
x1=0x_{1} = 0
- is an inflection point

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
[4.36777096705602,∞)\left[4.36777096705602, \infty\right)
Convex at the intervals
(−∞,4.36777096705602]\left(-\infty, 4.36777096705602\right]
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞x1x=1\lim_{x \to -\infty} x^{\frac{1}{x}} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞x1x=1\lim_{x \to \infty} x^{\frac{1}{x}} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x^(1/x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x1xx)=0\lim_{x \to -\infty}\left(\frac{x^{\frac{1}{x}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(x1xx)=0\lim_{x \to \infty}\left(\frac{x^{\frac{1}{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:
x1x=(−x)−1xx^{\frac{1}{x}} = \left(- x\right)^{- \frac{1}{x}}
- No
x1x=−(−x)−1xx^{\frac{1}{x}} = - \left(- x\right)^{- \frac{1}{x}}
- No
so, the function
not is
neither even, nor odd
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