Mister Exam
Graphing y = arccos*(2x-1)/(3)^1/2
The solution
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acos(2*x - 1)
f(x) = -------------
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$$f{\left(x \right)} = \frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}}$$
f = acos(2*x - 1)/sqrt(3)
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:
$$\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$- \frac{2 \frac{\sqrt{3}}{3}}{\sqrt{1 - \left(2 x - 1\right)^{2}}} = 0$$
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
$$- \frac{2 \frac{\sqrt{3}}{3}}{\sqrt{1 - \left(2 x - 1\right)^{2}}} = 0$$
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
$$- \frac{4 \sqrt{3} \left(2 x - 1\right)}{3 \left(1 - \left(2 x - 1\right)^{2}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{1}{2}$$
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:
Concave at the intervals
$$\left(-\infty, \frac{1}{2}\right]$$
Convex at the intervals
$$\left[\frac{1}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{4 \sqrt{3} \left(2 x - 1\right)}{3 \left(1 - \left(2 x - 1\right)^{2}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{1}{2}$$
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:
Concave at the intervals
$$\left(-\infty, \frac{1}{2}\right]$$
Convex at the intervals
$$\left[\frac{1}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}}\right) = \infty i$$
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 acos(2*x - 1)/sqrt(3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{\sqrt{3}}{3} \operatorname{acos}{\left(2 x - 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{\sqrt{3}}{3} \operatorname{acos}{\left(2 x - 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\frac{\sqrt{3}}{3} \operatorname{acos}{\left(2 x - 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{\sqrt{3}}{3} \operatorname{acos}{\left(2 x - 1 \right)}}{x}\right) = 0$$
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:
$$\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \operatorname{acos}{\left(- 2 x - 1 \right)}$$
- No
$$\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}} = - \frac{\sqrt{3}}{3} \operatorname{acos}{\left(- 2 x - 1 \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \operatorname{acos}{\left(- 2 x - 1 \right)}$$
- No
$$\frac{\operatorname{acos}{\left(2 x - 1 \right)}}{\sqrt{3}} = - \frac{\sqrt{3}}{3} \operatorname{acos}{\left(- 2 x - 1 \right)}$$
- No
so, the function
not is
neither even, nor odd