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  • Graphing y =:
  • x^3+4x
  • -x^2(x+4)^2
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  • Identical expressions

  • x^((one -log(x))/x)
  • x to the power of ((1 minus logarithm of (x)) divide by x)
  • x to the power of ((one minus logarithm of (x)) divide by x)
  • x((1-log(x))/x)
  • x1-logx/x
  • x^1-logx/x
  • x^((1-log(x)) divide by x)

  • Similar expressions

  • x^((1+log(x))/x)
Plotting a function graph/ x^((1-log(x))/x)

Graphing y = x^((1-log(x))/x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        1 - log(x)
        ----------
            x     
f(x) = x
f(x)=x1−log⁡(x)xf{\left(x \right)} = x^{\frac{1 - \log{\left(x \right)}}{x}} 
f = x^((1 - log(x))/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:
x1−log⁡(x)x=0x^{\frac{1 - \log{\left(x \right)}}{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 - log(x))/x).
01−log⁡(0)00^{\frac{1 - \log{\left(0 \right)}}{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
x1−log⁡(x)x((−1−log⁡(x)x2−1x2)log⁡(x)+1−log⁡(x)x2)=0x^{\frac{1 - \log{\left(x \right)}}{x}} \left(\left(- \frac{1 - \log{\left(x \right)}}{x^{2}} - \frac{1}{x^{2}}\right) \log{\left(x \right)} + \frac{1 - \log{\left(x \right)}}{x^{2}}\right) = 0
Solve this equation
The roots of this equation
x1=e32−52x_{1} = e^{\frac{3}{2} - \frac{\sqrt{5}}{2}}
x2=e52+32x_{2} = e^{\frac{\sqrt{5}}{2} + \frac{3}{2}}
The values of the extrema at the points:
                                                 ___ 
                                           3   \/ 5  
        ___   /        ___\ /      ___\  - - + ----- 
  3   \/ 5    |  1   \/ 5 | |3   \/ 5 |    2     2   
  - - -----   |- - + -----|*|- - -----|*e            
  2     2     \  2     2  / \2     2  /              
(e        , e                                      )

                                                 ___ 
                                           3   \/ 5  
        ___   /        ___\ /      ___\  - - - ----- 
  3   \/ 5    |  1   \/ 5 | |3   \/ 5 |    2     2   
  - + -----   |- - - -----|*|- + -----|*e            
  2     2     \  2     2  / \2     2  /              
(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:
Minima of the function at points:
x1=e52+32x_{1} = e^{\frac{\sqrt{5}}{2} + \frac{3}{2}}
Maxima of the function at points:
x1=e32−52x_{1} = e^{\frac{3}{2} - \frac{\sqrt{5}}{2}}
Decreasing at intervals
(−∞,e32−52]∪[e52+32,∞)\left(-\infty, e^{\frac{3}{2} - \frac{\sqrt{5}}{2}}\right] \cup \left[e^{\frac{\sqrt{5}}{2} + \frac{3}{2}}, \infty\right)
Increasing at intervals
[e32−52,e52+32]\left[e^{\frac{3}{2} - \frac{\sqrt{5}}{2}}, e^{\frac{\sqrt{5}}{2} + \frac{3}{2}}\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−log⁡(x)−1x(−(2log⁡(x)−5)log⁡(x)+3log⁡(x)−5+(−(log⁡(x)−2)log⁡(x)+log⁡(x)−1)2x)x3=0\frac{x^{- \frac{\log{\left(x \right)} - 1}{x}} \left(- \left(2 \log{\left(x \right)} - 5\right) \log{\left(x \right)} + 3 \log{\left(x \right)} - 5 + \frac{\left(- \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=77931.734784807x_{1} = 77931.734784807
x2=102935.111650972x_{2} = 102935.111650972
x3=97150.494187098x_{3} = 97150.494187098
x4=95722.5225642332x_{4} = 95722.5225642332
x5=25.8035786397959x_{5} = 25.8035786397959
x6=94304.3504508803x_{6} = 94304.3504508803
x7=108794.49750591x_{7} = 108794.49750591
x8=87961.4823832922x_{8} = 87961.4823832922
x9=100030.394442753x_{9} = 100030.394442753
x10=104394.661040139x_{10} = 104394.661040139
x11=78739.57382288x_{11} = 78739.57382288
x12=105858.105008557x_{12} = 105858.105008557
x13=80261.0338703568x_{13} = 80261.0338703568
x14=91504.4658766876x_{14} = 91504.4658766876
x15=81303.7826563622x_{15} = 81303.7826563622
x16=101480.105777427x_{16} = 101480.105777427
x17=107324.880569688x_{17} = 107324.880569688
x18=84848.3886498729x_{18} = 84848.3886498729
x19=88768.8309162064x_{19} = 88768.8309162064
x20=98586.8506610603x_{20} = 98586.8506610603
x21=77784.2394803298x_{21} = 77784.2394803298
x22=77627.743626596x_{22} = 77627.743626596
x23=83612.7146571421x_{23} = 83612.7146571421
x24=79323.6060511737x_{24} = 79323.6060511737
x25=90127.2011741679x_{25} = 90127.2011741679
x26=92897.6599631321x_{26} = 92897.6599631321
x27=86124.2767724825x_{27} = 86124.2767724825
x28=82426.889855648x_{28} = 82426.889855648
x29=78528.2363867842x_{29} = 78528.2363867842
x30=87433.0062814698x_{30} = 87433.0062814698
x31=2.08138681592069x_{31} = 2.08138681592069
x32=111724.750004227x_{32} = 111724.750004227
x33=81299.2959882933x_{33} = 81299.2959882933
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−(x−log⁡(x)−1x(−(2log⁡(x)−5)log⁡(x)+3log⁡(x)−5+(−(log⁡(x)−2)log⁡(x)+log⁡(x)−1)2x)x3)=∞\lim_{x \to 0^-}\left(\frac{x^{- \frac{\log{\left(x \right)} - 1}{x}} \left(- \left(2 \log{\left(x \right)} - 5\right) \log{\left(x \right)} + 3 \log{\left(x \right)} - 5 + \frac{\left(- \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}}\right) = \infty
lim⁡x→0+(x−log⁡(x)−1x(−(2log⁡(x)−5)log⁡(x)+3log⁡(x)−5+(−(log⁡(x)−2)log⁡(x)+log⁡(x)−1)2x)x3)=0\lim_{x \to 0^+}\left(\frac{x^{- \frac{\log{\left(x \right)} - 1}{x}} \left(- \left(2 \log{\left(x \right)} - 5\right) \log{\left(x \right)} + 3 \log{\left(x \right)} - 5 + \frac{\left(- \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \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
[2.08138681592069,25.8035786397959]\left[2.08138681592069, 25.8035786397959\right]
Convex at the intervals
(−∞,2.08138681592069]∪[25.8035786397959,∞)\left(-\infty, 2.08138681592069\right] \cup \left[25.8035786397959, \infty\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→−∞x1−log⁡(x)x=1\lim_{x \to -\infty} x^{\frac{1 - \log{\left(x \right)}}{x}} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞x1−log⁡(x)x=1\lim_{x \to \infty} x^{\frac{1 - \log{\left(x \right)}}{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 - log(x))/x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x1−log⁡(x)xx)=0\lim_{x \to -\infty}\left(\frac{x^{\frac{1 - \log{\left(x \right)}}{x}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(x1−log⁡(x)xx)=0\lim_{x \to \infty}\left(\frac{x^{\frac{1 - \log{\left(x \right)}}{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:
x1−log⁡(x)x=(−x)−1−log⁡(−x)xx^{\frac{1 - \log{\left(x \right)}}{x}} = \left(- x\right)^{- \frac{1 - \log{\left(- x \right)}}{x}}
- No
x1−log⁡(x)x=−(−x)−1−log⁡(−x)xx^{\frac{1 - \log{\left(x \right)}}{x}} = - \left(- x\right)^{- \frac{1 - \log{\left(- x \right)}}{x}}
- No
so, the function
not is
neither even, nor odd
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