Question

Using the completing-the-square method, find the vertex of the function f(x) = –2x^2 + 12x + 5 and indicate whether it is a minimum or a maximum and at what point.

A. Maximum at (–3, 5)

B. Minimum at (–3, 5)

C. Maximum at (3, 23)

D. Minimum at (3, 23)

Answer 1

Option C. Maximum at
`(3,23)`

Explanation:

we have

`f(x)=-2x^(2)+12x+5`

Completing the square

`f(x)-5=-2x^(2)+12x`

`f(x)-5=-2(x^(2)-6x)`

`f(x)-5-18=-2(x^(2)-6x+9)`

`f(x)-23=-2(x^(2)-6x+9)`

`f(x)-23=-2(x-3)^(2)`

`f(x)=-2(x-3)^(2)+23` --------> quadratic equation in vertex form

The vertex is the point
`(3,23)`

This is a vertical parabola open downward

therefore

The vertex is a maximum