Chapter 6, Problem 16MCQ

To determine

Find the x -coordinates at which the relative maxima and minima occur.

Answer to Problem 16MCQ

The x -coordinates $-0.535$ and $1.868$ represent the relative maxima and minima respectively.

Explanation of Solution

Given:

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

Calculation:

Draw a table for the values of $f\left(x\right)$ with respect to $x$ .

$x$	$f\left(x\right)=x^3-2x^2-3x$	$\left(x,f\left(x\right)\right)$
$\begin{array}{l}-2\\ -1\\ -0.535\\ 1.868\\ 3\\ 4\end{array}$	$\begin{array}{l}-10\\ 0\\ 0.879\\ -6.06\\ 0\\ 20\end{array}$	$\begin{array}{l}\left(-2,-10\right)\\ \left(-1,0\right)\\ \left(-0.535,0.879\right)\\ \left(1.868,-6.06\right)\\ \left(3,0\right)\\ \left(4,20\right)\end{array}$

The graph of the function is given below.

  

In the graph, point $A$ is a relative maximum of the function since no other nearby points have a greater y-coordinate. The graph is increasing as it approaches $A$ and decreasing at it moves from $A$ .

In the graph, point $B$ is a relative minimum of the function since no other nearby points have a lesser y-coordinate. The graph is decreasing as it approaches $B$ and increasing at it moves from $B$ .

Hence the x -coordinates $-0.535$ and $1.868$ represent the relative maxima and minima respectively.