Chapter 2, Problem 6CT

Graph the function $f(x)=-x^4+2x^3+4x^2-2$ on the interval $(-5,5)$ using a graphing utility. Then approximate any local maximum and local minimum values rounded to two decimal places. Determine where the function is increasing and where it is decreasing.

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 $(-5,5)$.

Explanation:

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 $(2.351,5)$.