Mister Exam
Graphing y = pi/2-arcsin(absolute((2x-1)/3))
The solution
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pi /|2*x - 1|\
f(x) = -- - asin||-------||
2 \| 3 |/
$$f{\left(x \right)} = - \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2}$$
f = -asin(Abs((2*x - 1)/3)) + pi/2
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:
$$- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
$$x_{2} = 2$$
Numerical solution
$$x_{1} = -1$$
$$x_{2} = 2$$
so we need to solve the equation:
$$- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
$$x_{2} = 2$$
Numerical solution
$$x_{1} = -1$$
$$x_{2} = 2$$
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
$$- \frac{2 \operatorname{sign}{\left(\frac{2 x}{3} - \frac{1}{3} \right)}}{3 \sqrt{1 - \frac{\left(2 x - 1\right)^{2}}{9}}} = 0$$
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
$$- \frac{2 \operatorname{sign}{\left(\frac{2 x}{3} - \frac{1}{3} \right)}}{3 \sqrt{1 - \frac{\left(2 x - 1\right)^{2}}{9}}} = 0$$
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
$$- \frac{4 \left(6 \delta\left(\frac{2 x - 1}{3}\right) + \frac{\left(2 x - 1\right) \operatorname{sign}{\left(2 x - 1 \right)}}{1 - \frac{\left(2 x - 1\right)^{2}}{9}}\right)}{27 \sqrt{1 - \frac{\left(2 x - 1\right)^{2}}{9}}} = 0$$
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
$$- \frac{4 \left(6 \delta\left(\frac{2 x - 1}{3}\right) + \frac{\left(2 x - 1\right) \operatorname{sign}{\left(2 x - 1 \right)}}{1 - \frac{\left(2 x - 1\right)^{2}}{9}}\right)}{27 \sqrt{1 - \frac{\left(2 x - 1\right)^{2}}{9}}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of pi/2 - asin(Abs((2*x - 1)/3)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2}}{x}\right)$$
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:
$$- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2} = - \operatorname{asin}{\left(\left|{\frac{2 x}{3} + \frac{1}{3}}\right| \right)} + \frac{\pi}{2}$$
- No
$$- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2} = \operatorname{asin}{\left(\left|{\frac{2 x}{3} + \frac{1}{3}}\right| \right)} - \frac{\pi}{2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2} = - \operatorname{asin}{\left(\left|{\frac{2 x}{3} + \frac{1}{3}}\right| \right)} + \frac{\pi}{2}$$
- No
$$- \operatorname{asin}{\left(\left|{\frac{2 x - 1}{3}}\right| \right)} + \frac{\pi}{2} = \operatorname{asin}{\left(\left|{\frac{2 x}{3} + \frac{1}{3}}\right| \right)} - \frac{\pi}{2}$$
- No
so, the function
not is
neither even, nor odd