Mister Exam
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-
How to use it?
- Graphing y =:
- x^2+3x+3
- -x^2-2x+1
- (x+1)(x-2)^2
- ((x-1)/(x+1))^3
- Derivative of:
-
acos(2*x)^(3)
-
Identical expressions
- acos(two *x)^(three)
- arc co sinus of e of ine of (2 multiply by x) to the power of (3)
- arc co sinus of e of ine of (two multiply by x) to the power of (three)
- acos(2*x)(3)
- acos2*x3
- acos(2x)^(3)
- acos(2x)(3)
- acos2x3
- acos2x^3
-
Similar expressions
- arccos(2*x)^(3)
Graphing y = acos(2*x)^(3)
The solution
You have entered
[src]
3 f(x) = acos (2*x)
$$f{\left(x \right)} = \operatorname{acos}^{3}{\left(2 x \right)}$$
f = acos(2*x)^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:
$$\operatorname{acos}^{3}{\left(2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
Numerical solution
$$x_{1} = 0.499999999901349$$
$$x_{2} = 0.500000002241688$$
$$x_{3} = 0.500000000882208$$
$$x_{4} = 0.499999999961417$$
$$x_{5} = 0.500000000037309$$
$$x_{6} = 0.500000000466316$$
so we need to solve the equation:
$$\operatorname{acos}^{3}{\left(2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
Numerical solution
$$x_{1} = 0.499999999901349$$
$$x_{2} = 0.500000002241688$$
$$x_{3} = 0.500000000882208$$
$$x_{4} = 0.499999999961417$$
$$x_{5} = 0.500000000037309$$
$$x_{6} = 0.500000000466316$$
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
$$- \frac{6 \operatorname{acos}^{2}{\left(2 x \right)}}{\sqrt{1 - 4 x^{2}}} = 0$$
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
$$- \frac{6 \operatorname{acos}^{2}{\left(2 x \right)}}{\sqrt{1 - 4 x^{2}}} = 0$$
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
$$- 24 \left(\frac{x \operatorname{acos}{\left(2 x \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}} + \frac{1}{4 x^{2} - 1}\right) \operatorname{acos}{\left(2 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 1.73358351657812$$
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
$$- 24 \left(\frac{x \operatorname{acos}{\left(2 x \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}} + \frac{1}{4 x^{2} - 1}\right) \operatorname{acos}{\left(2 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 1.73358351657812$$
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
$$\lim_{x \to -\infty} \operatorname{acos}^{3}{\left(2 x \right)} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \operatorname{acos}^{3}{\left(2 x \right)} = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \operatorname{acos}^{3}{\left(2 x \right)} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \operatorname{acos}^{3}{\left(2 x \right)} = - \infty i$$
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)^3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acos}^{3}{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{acos}^{3}{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acos}^{3}{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{acos}^{3}{\left(2 x \right)}}{x}\right) = 0$$
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:
$$\operatorname{acos}^{3}{\left(2 x \right)} = \operatorname{acos}^{3}{\left(- 2 x \right)}$$
- No
$$\operatorname{acos}^{3}{\left(2 x \right)} = - \operatorname{acos}^{3}{\left(- 2 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{acos}^{3}{\left(2 x \right)} = \operatorname{acos}^{3}{\left(- 2 x \right)}$$
- No
$$\operatorname{acos}^{3}{\left(2 x \right)} = - \operatorname{acos}^{3}{\left(- 2 x \right)}$$
- No
so, the function
not is
neither even, nor odd