Question

answer the questions below about the quadratic function.\\( f(x) = -2x^2 + 12x - 16 \\)\
(a) does the function have a minimum or maximum value?\
\\( \circ \\) minimum \\( \circ \\) maximum\
(b) where does the minimum or maximum value occur?\
\\( x = \square \\)\
(c) what is the functions minimum or maximum value?\
\\( \square \\)

Explanation:

Step1: Identify leading coefficient sign

For $f(x)=-2x^2+12x-16$, the leading coefficient $a=-2<0$. A negative leading coefficient means the parabola opens downward, so the function has a maximum value.

Step2: Find x-coordinate of vertex

Use vertex formula $x=-\frac{b}{2a}$. Here $a=-2$, $b=12$.
$x=-\frac{12}{2\times(-2)} = \frac{-12}{-4}=3$

Step3: Calculate maximum function value

Substitute $x=3$ into $f(x)$:
$f(3)=-2(3)^2+12(3)-16=-2\times9+36-16=-18+36-16=2$

Answer:

(a) Maximum
(b) $x=3$
(c) $2$