Answer
(a.) Maximum.
(b.) At $x=-3$, maximum value is $21$.
(c.) Domain $=(-\infty,\infty)$.
Range $=(-\infty,21]$.
Work Step by Step
The given function is a quadratic function:
$f(x)=-2x^2-12x+3$
The standard form of the quadratic function is
$f(x)=ax^2+bx+c$
$a=-2,b=-12$ and $c=3$
(a.)
Because $a<0$, the function has a maximum value.
(b.)
$x-$coordinate at which maximum value occurs is
$x=-\frac{b}{2a}$.
Substitute all values.
$x=-\frac{(-12)}{2(-2)}$.
Simplify.
$x=-3$.
Substitute the value of $x$ into the given function.
$f(-3)=-2(-3)^2-12(-3)+3$
Simplify.
$f(-3)=-18+36+3$
$f(-3)=21$
(c.)
Domain is all possible input values.
Domain $=(-\infty,\infty)$.
Range is all possible output values.
Maximum value is $21$.
Range $=(-\infty,21]$.
