To analyze the behavior of the function [tex]\( f(x) = -(x+6)(x+2) \)[/tex], we need to understand its properties, specifically where it increases and where it decreases.
1. Identify the roots: The given function is a quadratic function, which can be written in standard form as:
[tex]\[
f(x) = -(x+6)(x+2)
\][/tex]
Expanding this, we get:
[tex]\[
f(x) = -(x^2 + 8x + 12) = -x^2 - 8x - 12
\][/tex]
The roots of the function are [tex]\( x = -6 \)[/tex] and [tex]\( x = -2 \)[/tex].
2. Vertex of the parabola: Since this is a quadratic function that opens downwards (the coefficient of [tex]\( x^2 \)[/tex] is negative), the vertex represents the maximum point. The x-coordinate of the vertex can be found using the midpoint of the roots:
[tex]\[
x_{\text{vertex}} = \frac{x_1 + x_2}{2} = \frac{-6 + (-2)}{2} = \frac{-8}{2} = -4
\][/tex]
3. Intervals of increase and decrease:
- For [tex]\( x < -4 \)[/tex], the function is decreasing because we are moving away from the vertex to the left.
- For [tex]\( x > -4 \)[/tex], the function is also decreasing because we are moving away from the vertex to the right.
- Specifically, the function will be increasing as we move towards the vertex from each side. Therefore, it increases on the interval [tex]\( -6 < x < -2 \)[/tex].
Given this analysis, the correct statement about the function is:
[tex]\[ \text{The function is increasing for all real values of } x \text{ where } -6 < x < -2. \][/tex]
Thus, the answer is:
The function is increasing for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex].
