Chapter 1.1, Problem 36E

Solution

Tell whether the function has a minimum value or a maximum value. Then find the minimum value or a maximum value.

The function has a maximum value.

Maximum Value $=(3,22)$

Given:

The given function $g(x)=-3x^2+18x-5$

Concept Used:

• If the coefficient of $x^2$ is positive, the function has a minimum. 
• If the coefficient of $x^2$ is negative, the function has a maximum

Calculation:

The given function is $g(x)=-3x^2+18x-5$ .

  $a=-3,b=18,c=-5$

If $a<0$ maximum value (graph open down).

If $a>0$ minimum value (graph open up).

Here $a=-3$

  $\begin{array}{l}-3<0\\ a<0\end{array}$

So, the function has a maximum value.

Maximum value $=\left(\frac{-b}{2a},y(\frac{-b}{2a})\right)$

  $\begin{array}{c}(x)=\frac{-b}{2a}\\ =\frac{-(18)}{2(-3)}\\ =\frac{-18}{-6}\\ =3\end{array}$

  $\frac{-b}{2a}=3$

  $\begin{array}{c}g(\frac{-b}{2a})=g(3)\\ g(x)=-3x^2+18x-5\\ $ =-3{(3)}^2+18(3)-5$=-27+54-5\\ =-32+54\\ =22\end{array}$

So, Maximum value $=\left(\frac{-b}{2a},y(\frac{-b}{2a})\right)$

  $=(3,22)$