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Chapter 9: Problem 32

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4 / 3}\)

Short Answer

Expert verified
The graph of the function \(y=x^{4 / 3}\) has an inflection point at \(x = 0\) but does not have any relative extrema.

Step by step solution

01

Graph the function

Input the function \(y=x^{4 / 3}\) into the graphing utility. You can use any utility that suits your needs but a popular option is desmos.com, a free online graphing calculator. With the function inputted, a graph will be generated
02

Identify inflection point

An inflection point occurs where the curve changes concavity. For the function \(y=x^{4 / 3}\), the graph changes from being concave down to being concave up at \(x=0\). This is the only inflection point present.
03

Confirm absence of relative extrema

An extrema occurs at the high or low points of the curve. For \(y=x^{4 / 3}\), there are no extrema because the function increases for both \(x > 0\) and \(x < 0\). Because there are no maximum or minimum points, there are no points of relative extrema.

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Most popular questions from this chapter

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4}-8 x^{3}+18 x^{2}-16 x+5\)

The concentration \(C\) (in milligrams per milliliter) of a drug in a patient's bloodstream \(t\) hours after injection into muscle tissue is modeled by $$ C=\frac{3 t}{27+t^{3}} $$ Use differentials to approximate the change in the concentration when \(t\) changes from \(t=1\) to \(t=1.5\).

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Sketch a graph of a function \(f\) having the given characteristics. (There are many correct answers.) $$ \begin{aligned} &f(-1)=f(3)=0\\\ &f^{\prime}(1) \text { is undefined. }\\\ &f^{\prime}(x)<0 \text { if } x<1\\\ &f^{\prime}(x)>0 \text { if } x>1\\\ &f^{\prime \prime}(x)<0, x \neq 1\\\ &\lim _{x \rightarrow \infty} f(x)=4 \end{aligned} $$

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-4 x^{3}+6 x^{2}\)
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