Chapter 4, Problem 39CR

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 39CR

The x -intercepts, local maximum, local minimums, the increasing intervals and the decreasing intervals are 0.249 and 1.345, $\left(-1.133,7.065\right)$ , $\left(-2,6\right)$ and $\left(0.883,-3.17\right)$ , $-2<x<-1.133$ , $x>0.883$ and $x<-2$ and $-1.133<x<0.883$ respectively.

Explanation of Solution

Given information:

The function: $f\left(x\right)=x^4+3x^3-x^2-8x+2$

Calculation:

The given function is $f\left(x\right)=x^4+3x^3-x^2-8x+2$

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

$x$	1	0	1
$y$	$-3$	2	7

The graph of the above function is shown below:

  

From the graph, it is seen that the curve intersects the x -axis at the points $\left(0.249,0\right)$ and $\left(1.345,0\right)$ .

So, the x -intercepts are 0.249 and 1.345.

Also, the local minimums of the graph are $\left(-2,6\right)$ and $\left(0.883,-3.17\right)$ and the local maximum is $\left(-1.133,7.065\right)$ .

The function is increasing on the intervals $-2<x<-1.133$ and $x>0.883$ and is decreasing on the intervals $x<-2$ and $-1.133<x<0.883$ .

Hence, the x -intercepts, local maximum, local minimums, the increasing intervals and the decreasing intervals are 0.249 and 1.345, $\left(-1.133,7.065\right)$ , $\left(-2,6\right)$ and $\left(0.883,-3.17\right)$ , $-2<x<-1.133$ , $x>0.883$ and $x<-2$ and $-1.133<x<0.883$ respectively.