Mister Exam
Graphing y = arcsinx/(x^(3/4))
The solution
You have entered
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asin(x)
f(x) = -------
3/4
x
$$f{\left(x \right)} = \frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}}$$
f = asin(x)/x^(3/4)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{1}{x^{\frac{3}{4}} \sqrt{1 - x^{2}}} - \frac{3 \operatorname{asin}{\left(x \right)}}{4 x^{\frac{7}{4}}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{1}{x^{\frac{3}{4}} \sqrt{1 - x^{2}}} - \frac{3 \operatorname{asin}{\left(x \right)}}{4 x^{\frac{7}{4}}} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\sqrt[4]{x}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{3}{2 x^{\frac{7}{4}} \sqrt{1 - x^{2}}} + \frac{21 \operatorname{asin}{\left(x \right)}}{16 x^{\frac{11}{4}}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\sqrt[4]{x}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{3}{2 x^{\frac{7}{4}} \sqrt{1 - x^{2}}} + \frac{21 \operatorname{asin}{\left(x \right)}}{16 x^{\frac{11}{4}}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of asin(x)/x^(3/4), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}} x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}} x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}} x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}} x}\right)$$
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:
$$\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}} = - \frac{\operatorname{asin}{\left(x \right)}}{\left(- x\right)^{\frac{3}{4}}}$$
- No
$$\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}} = \frac{\operatorname{asin}{\left(x \right)}}{\left(- x\right)^{\frac{3}{4}}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}} = - \frac{\operatorname{asin}{\left(x \right)}}{\left(- x\right)^{\frac{3}{4}}}$$
- No
$$\frac{\operatorname{asin}{\left(x \right)}}{x^{\frac{3}{4}}} = \frac{\operatorname{asin}{\left(x \right)}}{\left(- x\right)^{\frac{3}{4}}}$$
- No
so, the function
not is
neither even, nor odd