Analyze and sketch a graph of the function. Label any

Mahagnazk 2021-11-23 Answered
Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results \(\displaystyle{y}={3}{x}^{{{\frac{{{2}}}{{{3}}}}}}-{2}{x}\)

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Expert Answer

Luis Sullivan
Answered 2021-11-24 Author has 1633 answers

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,
image
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.

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