Step 1 Given Function

\(\displaystyle{y}={3}{x}^{{{\frac{{{2}}}{{{3}}}}}}-{2}{x}\)

To Find: Relative extreme, Point of inflection and Asymptotes.

On Sketching the graph of given function we get,

Step 2: From first derivative test definition

Suppose that isa critical point of then

If \(\displaystyle{f}`{\left({x}\right)}{>}{0}\) to the left of \(\displaystyle{x}={c}\) and \(\displaystyle{f}`{\left({x}\right)}{<}{0}\) to the right of \(\displaystyle{x}={c}\) then \(\displaystyle{x}={c}\) is a local maximum.

If \(\displaystyle{f}`{\left({x}\right)}{<}{0}\) to the left of \(\displaystyle{x}={c}\) and f`(x)>0 to the right of \(\displaystyle{x}={c}\) then \(\displaystyle{x}={c}\) is a local minimum.

If f`(x) is the same sign on both sides of \(\displaystyle{x}={c}\) then \(\displaystyle{x}={c}\) is neither a local maximum nor a local local minimum.

Step 3: On differentiating the given equation we obtain,

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={3}\times{\left\lbrace{\frac{{{2}}}{{{3}}}}\right\rbrace}{x}^{{-{\frac{{{1}}}{{{3}}}}}}-{2}\)

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\left\lbrace{\frac{{{2}}}{{{x}^{{{\frac{{{1}}}{{{3}}}}}}}}}\right\rbrace}-{2}\)

Now, to find critical points substitute,

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={0}\)

\(\displaystyle{\left\lbrace{\frac{{{2}}}{{{x}^{{{\frac{{{1}}}{{{3}}}}}}}}}\right\rbrace}-{2}={0}\)

\(\displaystyle{\left\lbrace{\frac{{{2}}}{{{x}^{{{\frac{{{1}}}{{{3}}}}}}}}}\right\rbrace}={2}\)

\(\displaystyle{x}^{{{\frac{{{1}}}{{{3}}}}}}={1}\)

\(\displaystyle{x}={1}\)

So the critical points obtained

\(\displaystyle{x}={O}\) and \(\displaystyle{x}={1}\)

\(-\propto\) So the intervals are

Checking the sign of f`(x)

at each monotone interval we have,

Step 5: By the Inflection point Definition

An inflection point is a point on graph at which the second derivative changes sign

If \(\displaystyle{f}\text{}{\left({x}\right)}{>}{0}\) then f(x) concave upwards

IF \(\displaystyle{f}\text{}{\left({x}\right)}{<}{0}\) then f(x) concave downwards

Here, We have,

\(\displaystyle{f}\text{}{\left({x}\right)}=-{\frac{{{2}}}{{{3}{x}^{{\frac{{{4}}}{{{3}}}}}}}}\)

Checking the sign we obtain,

\(\begin{array}{|c|c|}\hline &-\propto<x<0&x=0&0<x<\propto\\ \hline Sign&-&NA&+\\ \hline Behavior&Concave Downward&NA&Concave Downward\\ \hline \end{array}\)

On the Above analysis we find that there are

No any point of Inflection that we have for the given function.

Resulting in No any Vertical as well as Horizontal Asymptotes.