Chapter 2, Problem 6CT

To determine

The function $f(x)=-x^4+2x^3+4x^2-2$ over the interval $(-5,5)$, using graphing utility, approximate local maximum and minimum values rounded up to two decimals, determine where the function is increasing and where it is decreasing.

Answer to Problem 6CT

Solution:

The graph of function $f(x)=-x^4+2x^3+4x^2-2$ is:

  

It has local maximum value $\left(2.35,15.55\right)$ .

It has local minimum value $\left(0,-2\right)$ .

It is increasing over the intervals $(-5,0.851)$ and $(0,2.351)$ .

It is decreasing over the interval $(-0.851,0)$ and $(2.351,5)$ .

Explanation of Solution

Given information:

The function is $f(x)=-x^4+2x^3+4x^2-2$ over the interval $(-5,5)$ .

To graph the function $f(x)=-x^4+2x^3+4x^2-2$ using graphing utility, use the below steps:

Step I: Press the ON key.

Step II: Now, press [Y=]. Input the right hand side of the function $f(x)=-x^4+2x^3+4x^2-2$ in $Y_1$ .

Step III: Press [WINDOW] key and set the viewing window as below:

  $\begin{array}{ll}X\min =-5\hfill & X\max =5\hfill \\ Y\min =-800\hfill & Y\max =20\hfill \end{array}$

Step IV: Then hit [Graph] key to view the graph.

The graph of the function is as follows:

  

To find local maximum and local minimum on graph using graphing utility, use below steps:

Step IV: Press [2ND][TRACE] to access the calculate menu.

Step V: Press [MAXIMUM] and press [ENTER].

Step VI: Set left bound by using left and right arrow. Click [ENTER].

Step VII: Set right bound by using left and right arrow. Click [ENTER].

Step VIII: Click [Enter] button twice.

It will give the maximum value $x=2.35078,y=15.5477$ .

Round it to two decimals $x=2.35,y=15.55$ .

Thus, the function has its local maximum value at $\left(2.35,15.55\right)$ .

To find local minimum value, use below steps:

Step IX: Press [2ND][TRACE] to access the calculate menu.

Step X: Press [MINIMUM] and press [ENTER].

Step XI: Set left bound by using left and right arrow. Click [ENTER].

Step XII: Set right bound by using left and right arrow. Click [ENTER].

Step XIII: Click [Enter] button twice.

It will give the minimum value $x=0.00000077,y=-2$ .

Round it to two decimals $x=0.00,y=-2$ .

Thus, the function has its local minimum value at $\left(0,-2\right)$ .

By observing the graph of function $f(x)=-x^4+2x^3+4x^2-2$, it is increasing over the intervals $(-5,0.851)$ and $(0,2.351)$ .

The function is decreasing over the interval $(-0.851,0)$ and $(2.351,5)$ .