Chapter 8.3, Problem 22E

To determine

To find :Whether the function has a minimum value or a maximum value.

Answer to Problem 22E

The given function has maximum value 12.

Explanation of Solution

Given function: $y=-5x^2+10x+7$

Calculation:

Since the coefficient of $x^2$ is negative, therefore, the given function has maximumvalue and no minimum value.

To find the maximum value of the function , we will complete the square as follows.

  $\begin{array}{l}y=-5x^2+10x+7\\ y=-5\left(x^2-2x\right)+7\\ y=-5\left(x^2-2x+1\right)+7+5\\ y=-5{\left(x-1\right)}^2+12\end{array}$

Since,

  $\begin{array}{l}{\left(x-1\right)}^2\ge 0\forall x\in R\\ -5{\left(x-1\right)}^2\le 0\forall x\in R\\ -5{\left(x-1\right)}^2+12\le 12\forall x\in R\end{array}$

Therefore, the given function has maximumvalue 12.