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

Answer:

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

Step-by-step 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