Answer
The function has a maximum value of $21$.
Work Step by Step
Let's compare $f(x)=-2x^2+12x+3$ to $f(x)=ax^2+bx+c$.
We can see that $a=-2$, $b=12$, $c=3$.
$a\lt0$, hence the graph opens down, thus its vertex is a maximum.
The maximum value is at $x=-\frac{b}{2a}=-\frac{12}{2\cdot(-2)}=3.$
Therefore the maximum value is
$f(3)=-2(3)^2+12(3)+3=21.$
