Let's analyze the quadratic function given:
[tex]\[ f(x) = -2x^2 - 16x - 30. \][/tex]
### Step-by-Step Solution:
1. Identify the Type of Quadratic Function:
- The quadratic term [tex]\(-2x^2\)[/tex] has a negative coefficient ([tex]\(-2\)[/tex]).
- When the coefficient of [tex]\(x^2\)[/tex] is negative, the parabola opens downward.
- Therefore, the function has a maximum value (because the vertex of the parabola is the highest point).
2. Find the Vertex of the Parabola:
- The vertex form of a quadratic function [tex]\(ax^2 + bx + c\)[/tex] gives us the x-coordinate of the vertex as:
[tex]\[ x = -\frac{b}{2a}. \][/tex]
- Here, [tex]\( a = -2 \)[/tex] and [tex]\( b = -16 \)[/tex].
- Plugging in these values:
[tex]\[ x = -\frac{-16}{2 \times -2} = \frac{16}{-4} = -4. \][/tex]
3. Calculate the Maximum Value of the Function:
- Now, substitute [tex]\( x = -4 \)[/tex] back into the original function to find the maximum value:
[tex]\[ f(-4) = -2(-4)^2 - 16(-4) - 30. \][/tex]
- Calculate each term:
[tex]\[ -2(-4)^2 = -2 \times 16 = -32, \][/tex]
[tex]\[ -16(-4) = 64, \][/tex]
[tex]\[ \text{constant term} = -30. \][/tex]
- Summing these results:
[tex]\[ f(-4) = -32 + 64 - 30 = 2. \][/tex]
### Summary:
- Does the function have a minimum or maximum value?
- Maximum
- What is the function's minimum or maximum value?
- The maximum value of the function is [tex]\( 2 \)[/tex].
- Where does the minimum or maximum value occur?
- The maximum value occurs at [tex]\( x = -4 \)[/tex].
So, the answers to the questions are:
1. Maximum
2. [tex]\(\boxed{2}\)[/tex]
3. [tex]\(x = \boxed{-4}\)[/tex]
