Mister Exam
Graphing y = (2/3)*arccos(x+1.5)
The solution
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2*acos(x + 3/2)
f(x) = ---------------
3
$$f{\left(x \right)} = \frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3}$$
f = 2*acos(x + 3/2)/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:
$$\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{1}{2}$$
Numerical solution
$$x_{1} = -0.5$$
so we need to solve the equation:
$$\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{1}{2}$$
Numerical solution
$$x_{1} = -0.5$$
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
$$- \frac{2}{3 \sqrt{1 - \left(x + \frac{3}{2}\right)^{2}}} = 0$$
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
$$- \frac{2}{3 \sqrt{1 - \left(x + \frac{3}{2}\right)^{2}}} = 0$$
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
$$- \frac{2 \left(x + \frac{3}{2}\right)}{3 \left(1 - \frac{\left(2 x + 3\right)^{2}}{4}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{3}{2}$$
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
$$\left(-\infty, - \frac{3}{2}\right]$$
Convex at the intervals
$$\left[- \frac{3}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{2 \left(x + \frac{3}{2}\right)}{3 \left(1 - \frac{\left(2 x + 3\right)^{2}}{4}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{3}{2}$$
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
$$\left(-\infty, - \frac{3}{2}\right]$$
Convex at the intervals
$$\left[- \frac{3}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3}\right) = \infty i$$
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 2*acos(x + 3/2)/3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3 x}\right) = 0$$
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:
$$\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3} = \frac{2 \operatorname{acos}{\left(\frac{3}{2} - x \right)}}{3}$$
- No
$$\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3} = - \frac{2 \operatorname{acos}{\left(\frac{3}{2} - x \right)}}{3}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3} = \frac{2 \operatorname{acos}{\left(\frac{3}{2} - x \right)}}{3}$$
- No
$$\frac{2 \operatorname{acos}{\left(x + \frac{3}{2} \right)}}{3} = - \frac{2 \operatorname{acos}{\left(\frac{3}{2} - x \right)}}{3}$$
- No
so, the function
not is
neither even, nor odd