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  • Identical expressions

  • asin(log(x))/x^ two
  • arc sinus of e of ( logarithm of (x)) divide by x squared
  • arc sinus of e of ( logarithm of (x)) divide by x to the power of two
  • asin(log(x))/x2
  • asinlogx/x2
  • asin(log(x))/x²
  • asin(log(x))/x to the power of 2
  • asinlogx/x^2
  • asin(log(x)) divide by x^2

  • Similar expressions

  • arcsin(log(x))/x^2
Plotting a function graph/ asin(log(x))/x^2

Graphing y = asin(log(x))/x^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       asin(log(x))
f(x) = ------------
             2     
            x
f(x)=asin⁡(log⁡(x))x2f{\left(x \right)} = \frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} 
f = asin(log(x))/x^2
The graph of the function
02468-8-6-4-2-10105-5
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:
asin⁡(log⁡(x))x2=0\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
Numerical solution
x1=1x_{1} = 1
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to asin(log(x))/x^2.
asin⁡(log⁡(0))02\frac{\operatorname{asin}{\left(\log{\left(0 \right)} \right)}}{0^{2}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
1xx21−log⁡(x)2−2asin⁡(log⁡(x))x3=0\frac{1}{x x^{2} \sqrt{1 - \log{\left(x \right)}^{2}}} - \frac{2 \operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{3}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
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
−1+log⁡(x)log⁡(x)2−11−log⁡(x)2+6asin⁡(log⁡(x))−41−log⁡(x)2x4=0\frac{- \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{\sqrt{1 - \log{\left(x \right)}^{2}}} + 6 \operatorname{asin}{\left(\log{\left(x \right)} \right)} - \frac{4}{\sqrt{1 - \log{\left(x \right)}^{2}}}}{x^{4}} = 0
Solve this equation
The roots of this equation
x1=2.09653484237811x_{1} = 2.09653484237811
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−(−1+log⁡(x)log⁡(x)2−11−log⁡(x)2+6asin⁡(log⁡(x))−41−log⁡(x)2x4)=∞i\lim_{x \to 0^-}\left(\frac{- \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{\sqrt{1 - \log{\left(x \right)}^{2}}} + 6 \operatorname{asin}{\left(\log{\left(x \right)} \right)} - \frac{4}{\sqrt{1 - \log{\left(x \right)}^{2}}}}{x^{4}}\right) = \infty i
lim⁡x→0+(−1+log⁡(x)log⁡(x)2−11−log⁡(x)2+6asin⁡(log⁡(x))−41−log⁡(x)2x4)=∞i\lim_{x \to 0^+}\left(\frac{- \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{\sqrt{1 - \log{\left(x \right)}^{2}}} + 6 \operatorname{asin}{\left(\log{\left(x \right)} \right)} - \frac{4}{\sqrt{1 - \log{\left(x \right)}^{2}}}}{x^{4}}\right) = \infty i
- limits are equal, then skip the corresponding 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
[2.09653484237811,∞)\left[2.09653484237811, \infty\right)
Convex at the intervals
(−∞,2.09653484237811]\left(-\infty, 2.09653484237811\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→−∞(asin⁡(log⁡(x))x2)=0\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(asin⁡(log⁡(x))x2)=0\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}}\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 asin(log(x))/x^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(asin⁡(log⁡(x))xx2)=0\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(asin⁡(log⁡(x))xx2)=0\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x x^{2}}\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:
asin⁡(log⁡(x))x2=asin⁡(log⁡(−x))x2\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} = \frac{\operatorname{asin}{\left(\log{\left(- x \right)} \right)}}{x^{2}}
- No
asin⁡(log⁡(x))x2=−asin⁡(log⁡(−x))x2\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} = - \frac{\operatorname{asin}{\left(\log{\left(- x \right)} \right)}}{x^{2}}
- No
so, the function
not is
neither even, nor odd
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