Let's analyze the given quadratic function:
[tex]\[ g(x) = -2x^2 - 4x - 4 \][/tex]
### Step-by-Step Solution
1. Determine if the function has a maximum or minimum value:
- The quadratic function is in the standard form [tex]\( g(x) = ax^2 + bx + c \)[/tex].
- Here, the coefficient of [tex]\( x^2 \)[/tex], which is [tex]\( a \)[/tex], is [tex]\(-2\)[/tex].
- Since [tex]\( a \)[/tex] is negative ([tex]\( a < 0 \)[/tex]), the parabola opens downwards.
- Therefore, the function has a maximum value.
2. Find the x-coordinate of the vertex where the maximum value occurs:
- The x-coordinate of the vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[
x = \frac{-b}{2a}
\][/tex]
- Substitute [tex]\( a = -2 \)[/tex] and [tex]\( b = -4 \)[/tex]:
[tex]\[
x = \frac{-(-4)}{2(-2)} = \frac{4}{-4} = -1
\][/tex]
3. Find the maximum value of the function:
- Substitute [tex]\( x = -1 \)[/tex] back into the quadratic function [tex]\( g(x) \)[/tex]:
[tex]\[
g(-1) = -2(-1)^2 - 4(-1) - 4
\][/tex]
- Calculate each term:
[tex]\[
g(-1) = -2(1) - 4(-1) - 4 = -2 + 4 - 4 = -2
\][/tex]
### Answers
Does the function have a minimum or maximum value?
- Maximum
Where does the minimum or maximum value occur?
[tex]\[ x = -1 \][/tex]
What is the function's minimum or maximum value?
[tex]\[ \boxed{-2} \][/tex]
Therefore, the function has a maximum value which occurs at [tex]\( x = -1 \)[/tex] and its maximum value is [tex]\(-2\)[/tex].
