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Graphing y = x^(2/3)

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        2/3
f(x) = x

f{\left(x \right)} = x^{\frac{2}{3}}

f = x^(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:

x^{\frac{2}{3}} = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = 0

Numerical solution

x_{1} = 0

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to x^(2/3).

0^{\frac{2}{3}}

The result:

f{\left(0 \right)} = 0

The point:

(0, 0)

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)} =

\frac{2}{3 \sqrt[3]{x}} = 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)} =

- \frac{2}{9 x^{\frac{4}{3}}} = 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} x^{\frac{2}{3}} = \infty \left(-1\right)^{\frac{2}{3}}

Let's take the limit
so,
equation of the horizontal asymptote on the left:

y = \infty \left(-1\right)^{\frac{2}{3}}

\lim_{x \to \infty} x^{\frac{2}{3}} = \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^(2/3), divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty} \frac{1}{\sqrt[3]{x}} = 0

Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right

\lim_{x \to \infty} \frac{1}{\sqrt[3]{x}} = 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:

x^{\frac{2}{3}} = \left(- x\right)^{\frac{2}{3}}

- No

x^{\frac{2}{3}} = - \left(- x\right)^{\frac{2}{3}}

- No
so, the function
not is
neither even, nor odd