Chapter 4, Problem 38CR

To determine

To graph: The function $f\left(x\right)$ . Also, determine the x -intercepts, local maximum, local minimum and the intervals for which the function is increasing or decreasing.

Answer to Problem 38CR

The x -intercept, local maximum, local minimum, the increasing interval and the decreasing intervals are $-1.68$ , $\left(0,-1\right)$ , $\left(-1,-2\right)$ , $\left(-1,0\right)$ and $\left(-\infty ,-1\right)\cup \left(0,\infty \right)$ respectively.

Explanation of Solution

Given information:

The function: $f\left(x\right)=-2x^3-3x^2-1$

Calculation:

The given function is $f\left(x\right)=-2x^3-3x^2-1$

To graph the above function, plot the points that satisfy the function on the table below:

$x$	$-1$	0	1
$y$	$-2$	$-1$	$-6$

The graph of the above function is shown below:

  

From the graph, it is seen that the curve intersects the x -axis at the point $\left(-1.68,0\right)$ .

So, the x -intercept is $-1.68$ .

Also, the local minimum of the graph is $\left(-1,-2\right)$ and the local maximum is $\left(0,-1\right)$ .

The function is increasing on the interval $\left(-1,0\right)$ and is decreasing on the intervals $\left(-\infty ,-1\right)\cup \left(0,\infty \right)$ .

Hence, the x -intercept, local maximum, local minimum, the increasing interval and the decreasing intervals are $-1.68$ , $\left(0,-1\right)$ , $\left(-1,-2\right)$ , $\left(-1,0\right)$ and $\left(-\infty ,-1\right)\cup \left(0,\infty \right)$ respectively.