Mister Exam
Graphing y = pi/4+2arcsin(1-(x/4))
The solution
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pi / x\
f(x) = -- + 2*asin|1 - -|
4 \ 4/
$$f{\left(x \right)} = 2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4}$$
f = 2*asin(-x/4 + 1) + pi/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:
$$2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 2 \sqrt{2 - \sqrt{2}} + 4$$
Numerical solution
$$x_{1} = 5.53073372946036$$
so we need to solve the equation:
$$2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 2 \sqrt{2 - \sqrt{2}} + 4$$
Numerical solution
$$x_{1} = 5.53073372946036$$
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
$$- \frac{1}{2 \sqrt{1 - \left(- \frac{x}{4} + 1\right)^{2}}} = 0$$
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
$$- \frac{1}{2 \sqrt{1 - \left(- \frac{x}{4} + 1\right)^{2}}} = 0$$
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
$$- \frac{x - 4}{32 \left(1 - \frac{\left(4 - x\right)^{2}}{16}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 4$$
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
$$\left(-\infty, 4\right]$$
Convex at the intervals
$$\left[4, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{x - 4}{32 \left(1 - \frac{\left(4 - x\right)^{2}}{16}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 4$$
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
$$\left(-\infty, 4\right]$$
Convex at the intervals
$$\left[4, \infty\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of pi/4 + 2*asin(1 - x/4), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4}}{x}\right)$$
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:
$$2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4} = 2 \operatorname{asin}{\left(\frac{x}{4} + 1 \right)} + \frac{\pi}{4}$$
- No
$$2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4} = - 2 \operatorname{asin}{\left(\frac{x}{4} + 1 \right)} - \frac{\pi}{4}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4} = 2 \operatorname{asin}{\left(\frac{x}{4} + 1 \right)} + \frac{\pi}{4}$$
- No
$$2 \operatorname{asin}{\left(- \frac{x}{4} + 1 \right)} + \frac{\pi}{4} = - 2 \operatorname{asin}{\left(\frac{x}{4} + 1 \right)} - \frac{\pi}{4}$$
- No
so, the function
not is
neither even, nor odd