Chapter 4.3, Problem 21E

(a)

To determine

To find:The intervals of increase or decrease.

Explanation of Solution

Given: $f\left(x\right)=2x^3-3x^2-12x$

Calculate the first derivative of the given function.

  $\begin{array}{c}{f}^{\prime }\left(x\right)=6x^2-6x-12\\ =6\left(x^2-x-2\right)\end{array}$

Equate the derivative to zero.

  $\begin{array}{c}{x}^2-x-2=0\\ \left(x-2\right)\left(x+1\right)=0\\ x=-1,2\end{array}$

So, the critical points are $-1,2$ .

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

The function is decreasing in the interval $\left(-1,2\right)$

The function is increasing in the interval $\left(2,\infty \right)$ .

(b)

To determine

To find :The local maxima and local minima.

Explanation of Solution

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

The function is decreasing in the interval $\left(-1,2\right)$

The function is increasing in the interval $\left(2,\infty \right)$ .

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

So, the local maxima is $\left(-1,f\left(-1\right)\right)=\left(-1,7\right)$

The value of $f^{\prime }$ is changing from negative to positive at $x=2$ .

So, the local minima is $\left(2,f\left(2\right)\right)=\left(2,-20\right)$

(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.

  $\begin{array}{c}{f}^{″}\left(x\right)=12x-6\\ 12x-6=0\\ x=\frac{1}{2}\end{array}$

So,

  $\begin{array}{c}f\left(\frac{1}{2}\right)=2{\left(\frac{1}{2}\right)}^3-3{\left(\frac{1}{2}\right)}^2-12\left(\frac{1}{2}\right)\\ =-\frac{13}{2}\end{array}$

  $f^{″}\left(x\right)<0$ for the interval $-\infty <x<\frac{1}{2}$ . So, the graph is concave down for $\left(-\infty ,\frac{1}{2}\right)$

  $f^{″}\left(x\right)>0$ for the interval $\frac{1}{2}<x<\infty$ . So, the graph is concave up for $\left(\frac{1}{2},\infty \right)$ .

The inflection point is $\left(\frac{1}{2},\frac{-13}{2}\right)$ .

(d)

To determine

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

Explanation of Solution

Given: $f\left(x\right)=2x^3-3x^2-12x$

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

  

Figure (1)

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