Chapter 3, Problem 7CT

To determine

The following has either maximum or minimum value is to be stated and then its value to be estimated:

  $f\left(x\right)=-2x^2+12x+3$

Answer to Problem 7CT

The maximum value is $21$ .

Explanation of Solution

Given:

  $f\left(x\right)=-2x^2+12x+3$

Calculation:

The quadratic equations are given as:

  $f\left(x\right)=-2x^2+12x+3$

Comapre it with the standard quadratic equation: $f\left(x\right)=a{x}^2+bx+c$

  $\begin{array}{l}a=-2\\ b=12\\ c=3\end{array}$

For determining whether the graph of the function opens up or down. The graph of a quadratic function $f\left(x\right)=a{x}^2+bx+c$ opens up if $a>0$ and opens down if $a<0$ .

Function has $a=-2$ which is less than $0$ .

Thus, here graph opens down.

For determining whether vertex s a minimum or maximum point.

If $a<0$ the vertex is a highest point on the graph. Also, if $a>0$ , then the vertex is a lowest point on the graph .

Since, $a<0$ , the vertex is a maximum point.

To find maximum value.

Maximum value of the function $f$ is $f\left(\frac{-b}{2a}\right)$ if the vertex is the highest point.

Substitute the $a=-2,b=12$ in $\frac{-b}{2a}$ .

Minimum value is at

  $\begin{array}{l}x=\frac{-b}{2a}\\ =\frac{-12}{2\left(-2\right)}\\ =3\end{array}$

For evaluating the function at $\frac{-b}{2a}$ by substituting $x=3$ in the function $f\left(x\right)=-2x^2+12x+3$ to find the maximum value,

  $\begin{array}{l}f\left(3\right)=-2{\left(3\right)}^2+12\left(3\right)+3\\ =-18+36+3\\ =21\end{array}$

Conclusion:

Hence, maximum value is $21$ .