Mister Exam
Graphing y = cbrt((x+1)^2)
The solution
You have entered
[src]
__________
3 / 2
f(x) = \/ (x + 1)
$$f{\left(x \right)} = \sqrt[3]{\left(x + 1\right)^{2}}$$
f = ((x + 1)^2)^(1/3)
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:
$$\sqrt[3]{\left(x + 1\right)^{2}} = 0$$
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:
$$\sqrt[3]{\left(x + 1\right)^{2}} = 0$$
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
$$\frac{\left(\frac{2 x}{3} + \frac{2}{3}\right) \left|{x + 1}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{2}} = 0$$
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
$$\frac{\left(\frac{2 x}{3} + \frac{2}{3}\right) \left|{x + 1}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{2}} = 0$$
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
$$\frac{2 \left(\frac{2 \operatorname{sign}{\left(x + 1 \right)}}{\sqrt[3]{\left|{x + 1}\right|}} - \frac{3 \left|{x + 1}\right|^{\frac{2}{3}}}{x + 1}\right)}{9 \left(x + 1\right)} = 0$$
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
$$\frac{2 \left(\frac{2 \operatorname{sign}{\left(x + 1 \right)}}{\sqrt[3]{\left|{x + 1}\right|}} - \frac{3 \left|{x + 1}\right|^{\frac{2}{3}}}{x + 1}\right)}{9 \left(x + 1\right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt[3]{\left(x + 1\right)^{2}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt[3]{\left(x + 1\right)^{2}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \sqrt[3]{\left(x + 1\right)^{2}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt[3]{\left(x + 1\right)^{2}} = \infty$$
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 ((x + 1)^2)^(1/3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 1}\right|^{\frac{2}{3}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{x + 1}\right|^{\frac{2}{3}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 1}\right|^{\frac{2}{3}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{x + 1}\right|^{\frac{2}{3}}}{x}\right) = 0$$
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:
$$\sqrt[3]{\left(x + 1\right)^{2}} = \left|{x - 1}\right|^{\frac{2}{3}}$$
- No
$$\sqrt[3]{\left(x + 1\right)^{2}} = - \left|{x - 1}\right|^{\frac{2}{3}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt[3]{\left(x + 1\right)^{2}} = \left|{x - 1}\right|^{\frac{2}{3}}$$
- No
$$\sqrt[3]{\left(x + 1\right)^{2}} = - \left|{x - 1}\right|^{\frac{2}{3}}$$
- No
so, the function
not is
neither even, nor odd