Question

Determine whether f(x) = –5x^2 – 10x + 6 has a maximum or a minimum value. Find that value and explain how you know.

Answer 1

The vertex is the point (-1,11). is a maximum

Explanation:

we know that

The equation of a vertical parabola in vertex form is equal to

`f(x)=a(x-h)^(2)+k`

where

(h,k) is the vertex

a is a coefficient

if a > 0 the parabola open upward and the vertex is a minimum

if a < 0 the parabola open downward and the vertex is a maximum

we have

`f(x)=-5x^(2)-10x+6`

Convert to vertex form

Complete the square

`f(x)-6=-5x^(2)-10x`

Factor the leading coefficient

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

`f(x)-6-5=-5(x^(2)+2x+1)`

`f(x)-11=-5(x^(2)+2x+1)`

Rewrite as perfect squares

`f(x)-11=-5(x+1)^(2)`

`f(x)=-5(x+1)^(2)+11`

The vertex is the point (-1,11)

The coefficient a=-5

so

a < 0 the parabola open downward and the vertex is a maximum