Question

consider the function $f(x) = -x^2 - 2x + 8$
select the interval of values for which $f(x)$ is positive and increasing.

Explanation:

Step1: Find roots of \( f(x) = 0 \)

Solve \( -x^2 - 2x + 8 = 0 \). Multiply by -1: \( x^2 + 2x - 8 = 0 \). Factor: \( (x + 4)(x - 2) = 0 \). Roots: \( x = -4, x = 2 \).

Step2: Analyze parabola direction

The coefficient of \( x^2 \) is -1 (negative), so parabola opens downward. \( f(x) > 0 \) between roots: \( (-4, 2) \).

Step3: Find derivative for increasing

\( f'(x) = -2x - 2 \). Set \( f'(x) > 0 \): \( -2x - 2 > 0 \Rightarrow -2x > 2 \Rightarrow x < -1 \).

Step4: Intersection of intervals

\( f(x) > 0 \) is \( (-4, 2) \), \( f(x) \) increasing is \( (-\infty, -1) \). Intersection: \( (-4, -1) \).

Answer:

The interval is \((-4, -1)\)