Mister Exam
Graphing y = asin(2*sqrt(x*(1-x)))/4-(1/2-x)*sqrt(x*(1-x))
The solution
You have entered
[src]
/ ___________\
asin\2*\/ x*(1 - x) / ___________
f(x) = --------------------- - (1/2 - x)*\/ x*(1 - x)
4
$$f{\left(x \right)} = - \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4}$$
f = -sqrt(x*(1 - x))*(1/2 - x) + asin(2*sqrt(x*(1 - x)))/4
The graph of the function
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:
$$- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
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
$$\sqrt{x \left(1 - x\right)} + \frac{\sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) \left(x - \frac{1}{2}\right)}{x \left(1 - x\right)} + \frac{\sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right)}{2 x \left(1 - x\right) \sqrt{- 4 x \left(1 - x\right) + 1}} = 0$$
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
$$\sqrt{x \left(1 - x\right)} + \frac{\sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) \left(x - \frac{1}{2}\right)}{x \left(1 - x\right)} + \frac{\sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right)}{2 x \left(1 - x\right) \sqrt{- 4 x \left(1 - x\right) + 1}} = 0$$
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
$$\frac{\sqrt{- x \left(x - 1\right)} \left(3 x - \frac{\left(2 x - 1\right)^{2}}{2 \left(- 4 x \left(1 - x\right) + 1\right)^{\frac{3}{2}}} - \frac{3}{2} + \frac{1}{2 \sqrt{- 4 x \left(1 - x\right) + 1}} - \frac{\left(2 x - 1\right)^{2}}{4 \left(x - 1\right)} - \frac{2 x - 1}{4 \left(x - 1\right) \sqrt{- 4 x \left(1 - x\right) + 1}} - \frac{\left(2 x - 1\right)^{2}}{4 x} - \frac{2 x - 1}{4 x \sqrt{- 4 x \left(1 - x\right) + 1}} + \frac{\left(2 x - 1\right)^{3}}{8 x \left(x - 1\right)} + \frac{\left(2 x - 1\right)^{2}}{8 x \left(x - 1\right) \sqrt{- 4 x \left(1 - x\right) + 1}}\right)}{x \left(x - 1\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{1}{2} + \frac{\sqrt{2}}{2}$$
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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\sqrt{- x \left(x - 1\right)} \left(3 x - \frac{\left(2 x - 1\right)^{2}}{2 \left(- 4 x \left(1 - x\right) + 1\right)^{\frac{3}{2}}} - \frac{3}{2} + \frac{1}{2 \sqrt{- 4 x \left(1 - x\right) + 1}} - \frac{\left(2 x - 1\right)^{2}}{4 \left(x - 1\right)} - \frac{2 x - 1}{4 \left(x - 1\right) \sqrt{- 4 x \left(1 - x\right) + 1}} - \frac{\left(2 x - 1\right)^{2}}{4 x} - \frac{2 x - 1}{4 x \sqrt{- 4 x \left(1 - x\right) + 1}} + \frac{\left(2 x - 1\right)^{3}}{8 x \left(x - 1\right)} + \frac{\left(2 x - 1\right)^{2}}{8 x \left(x - 1\right) \sqrt{- 4 x \left(1 - x\right) + 1}}\right)}{x \left(x - 1\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{1}{2} + \frac{\sqrt{2}}{2}$$
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:
Have no bends at the whole real axis
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:
$$y = \lim_{x \to -\infty}\left(- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4}\right)$$
$$\lim_{x \to \infty}\left(- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4}\right)$$
$$\lim_{x \to \infty}\left(- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4}\right) = \infty i$$
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 asin(2*sqrt(x*(1 - x)))/4 - (1/2 - x)*sqrt(x*(1 - x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4}}{x}\right)$$
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:
$$- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4} = - \sqrt{- x \left(x + 1\right)} \left(x + \frac{1}{2}\right) + \frac{\operatorname{asin}{\left(2 \sqrt{- x \left(x + 1\right)} \right)}}{4}$$
- No
$$- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4} = \sqrt{- x \left(x + 1\right)} \left(x + \frac{1}{2}\right) - \frac{\operatorname{asin}{\left(2 \sqrt{- x \left(x + 1\right)} \right)}}{4}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4} = - \sqrt{- x \left(x + 1\right)} \left(x + \frac{1}{2}\right) + \frac{\operatorname{asin}{\left(2 \sqrt{- x \left(x + 1\right)} \right)}}{4}$$
- No
$$- \sqrt{x \left(1 - x\right)} \left(\frac{1}{2} - x\right) + \frac{\operatorname{asin}{\left(2 \sqrt{x \left(1 - x\right)} \right)}}{4} = \sqrt{- x \left(x + 1\right)} \left(x + \frac{1}{2}\right) - \frac{\operatorname{asin}{\left(2 \sqrt{- x \left(x + 1\right)} \right)}}{4}$$
- No
so, the function
not is
neither even, nor odd