Chapter 4.3, Problem 29E

(a)

To determine

To find:The intervals of increase or decrease.

Explanation of Solution

Given: $C\left(x\right)=x^{1/3}\left(x+4\right)$

Calculate the first derivative of the given function.

  $\begin{array}{c}{C}^{\prime }\left(x\right)=\frac{1}{3x^{2/3}}\left(x+4\right)+x^{1/3}\\ =\frac{x+4+3x}{3x^{2/3}}\\ =\frac{4}{3}\left(\frac{x+1}{x^{2/3}}\right)\end{array}$

Equate to zero.

  $x=0,-1$

The function is decreasing in the interval $\left(-\infty ,1\right)$ .

The function is increasing in the interval $\left(-1,0\right)$

The function is decreasing in the interval $\left(0,\infty \right)$ .

(b)

To determine

To find :The local maxima and local minima.

Explanation of Solution

The function is decreasing in the interval $\left(-\infty ,1\right)$ .

The function is increasing in the interval $\left(-1,0\right)$

The function is decreasing in the interval $\left(0,\infty \right)$ .

The value of $C^{\prime }$ is changing from negative to positive at $x=-1$

So, the local minima is $\left(-1,-3\right)$

There is no local maxima.

(c)

To determine

To find :The concavity of the function and the point of inflection.

Explanation of Solution

Calculate the second derivative of the given function.

  $\begin{array}{c}{C}^{″}\left(x\right)=\frac{4\left(x+1\right)}{3x^{2/3}}\\ =\frac{4}{3}\left[\frac{x-2}{3x^{5/3}}\right]\\ =\frac{4}{9}\left[\frac{x-2}{x^{5/3}}\right]\end{array}$

  $C^{″}\left(x\right)>0$ for the interval $-\infty <x<0$ . So, the graph is concave up for $\left(-\infty ,0\right)$

  $C^{″}\left(x\right)<0$ for the interval $0<x<2$ . So, the graph is concave down for $\left(0,2\right)$

  $C^{″}\left(x\right)>0$ for the interval $2<x<\infty$ . So, the graph is concave up for $\left(2,\infty \right)$

The inflection points are $\left(0,0\right)\text{and}\left(2,6\sqrt[3]{2}\right)$

(d)

To determine

To sketch :The graph of result obtained from part (a) to (c).

Explanation of Solution

Given: $C\left(x\right)=x^{1/3}\left(x+4\right)$

The graph of the result obtained is shown in figure below.

  

Figure (1)

Therefore, the required graph is shown in figure (1).