Mister Exam
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-
How to use it?
- Graphing y =:
- x^2+3x+3
- -x^2-2x+1
- x^2+14x+15
- x^2-2lnx
- How do you in partial fractions?:
- 2/(3*x^(1/3))
- Limit of the function:
-
2/(3*x^(1/3))
-
Identical expressions
- two /(three *x^(one / three))
- 2 divide by (3 multiply by x to the power of (1 divide by 3))
- two divide by (three multiply by x to the power of (one divide by three))
- 2/(3*x(1/3))
- 2/3*x1/3
- 2/(3x^(1/3))
- 2/(3x(1/3))
- 2/3x1/3
- 2/3x^1/3
- 2 divide by (3*x^(1 divide by 3))
Graphing y = 2/(3*x^(1/3))
The solution
You have entered
[src]
2
f(x) = -------
3 ___
3*\/ x
$$f{\left(x \right)} = \frac{2}{3 \sqrt[3]{x}}$$
f = 2/((3*x^(1/3)))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{2}{3 \sqrt[3]{x}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{2}{3 \sqrt[3]{x}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$- \frac{2}{9 x^{\frac{4}{3}}} = 0$$
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
$$- \frac{2}{9 x^{\frac{4}{3}}} = 0$$
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
$$\frac{8}{27 x^{\frac{7}{3}}} = 0$$
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
$$\frac{8}{27 x^{\frac{7}{3}}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{2}{3 \sqrt[3]{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{2}{3 \sqrt[3]{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{2}{3 \sqrt[3]{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{2}{3 \sqrt[3]{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2/((3*x^(1/3))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2 \frac{1}{3 \sqrt[3]{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 \frac{1}{3 \sqrt[3]{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{2 \frac{1}{3 \sqrt[3]{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 \frac{1}{3 \sqrt[3]{x}}}{x}\right) = 0$$
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:
$$\frac{2}{3 \sqrt[3]{x}} = \frac{2}{3 \sqrt[3]{- x}}$$
- No
$$\frac{2}{3 \sqrt[3]{x}} = - \frac{2}{3 \sqrt[3]{- x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{2}{3 \sqrt[3]{x}} = \frac{2}{3 \sqrt[3]{- x}}$$
- No
$$\frac{2}{3 \sqrt[3]{x}} = - \frac{2}{3 \sqrt[3]{- x}}$$
- No
so, the function
not is
neither even, nor odd