Chapter 3.1, Problem 57E

To determine

To find: The local maximum and minimum values of the function $g\left(x\right)=x^4-2x^3-11x^2$ and also the values of $x$ on which it occurs.

Answer to Problem 57E

The local maximum value of the function $g\left(x\right)=x^4-2x^3-11x^2$ is equal to $0$ at $x=0$ and the local minimum value of the function $g\left(x\right)=x^4-2x^3-11x^2$ is equal to $-13.614$ at $x=-1.712$ and $-73.323$ at $x=3.212$ .

Explanation of Solution

Given information:

The given function is $g\left(x\right)=x^4-2x^3-11x^2$ .

Calculation:

Graph the given function $g\left(x\right)=x^4-2x^3-11x^2$ .

To graph a function $y=x^4-2x^3-11x^2$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=x^4-2x^3-11x^2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{^}\boxed{4}\boxed{-}\boxed{2}\boxed{\text{X}}\boxed{^}\boxed{3}\boxed{-}\boxed{11}\boxed{\text{X}}\boxed{^}\boxed{2}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\text{X}^4-2\text{X}^3-11\text{X}^2$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown in Figure (1).

  

Figure (1)

As observed from the graph, the function has local maximum value equal to $0$ at $x=0$ .

As observed from the graph, the function has local minima value equal to $-13.614$ at $x=-1.712$ and $-73.323$ at $x=3.212$ .

Therefore, the local maximum value of the function $g\left(x\right)=x^4-2x^3-11x^2$ is equal to $0$ at $x=0$ and the local minimum value of the function $g\left(x\right)=x^4-2x^3-11x^2$ is equal to $-13.614$ at $x=-1.712$ and $-73.323$ at $x=3.212$ .