Question

1. Does the function have a minimum or maximum value?

Answer 1

Given the function $g(x)=2x^2+12x+21$. The above function can be written as

$g(x)=2x^2+12x+21 \\ g(x)=2(x^2+6x)+21 \\ g(x)=2(x^2+6x+9-9)+21 \\ g(x)=2(x^2+6x+9)-18+21 \\ g(x)=2(x+3)^2+3$

a)Now, the function $g(x)=2(x+3)^2+3$ has minimum value since the coefficient of $(x+3)^2$ is $2>0$.

b) The minimum value of the function occurs at $x=-3$ and its value is

$g(-3)=3(-3+3)^2+3 =3$

c)The minimum value of the function occurs at $x=-3$.