Question

given the graph of the function f below, determine all intervals on the open interval (-9, 9) where f(x) ≥ 0 and f(x) ≤ 0. f(-8.5) = 3.6281 graph of f answer attempt 1 out of 2 additional solution no solution

Explanation:

Step1: Analyze \( f'(x) \geq 0 \) (increasing)

\( f'(x) \geq 0 \) means the function \( f(x) \) is non - decreasing (increasing or constant). From the graph, we look for intervals where the slope of the tangent is non - negative. We can see that on \( (-9, - 7) \) (the left - most part of the graph) and on \( (4, 7) \) (the part of the graph that is rising from \( x = 4 \) to \( x=7 \)) the function is increasing (or non - decreasing).

Step2: Analyze \( f(x)\leq0 \) (below or on x - axis)

\( f(x)\leq0 \) means the graph of the function is below or on the x - axis. From the graph, we see that \( f(x)\leq0 \) on \( (-9, - 5)\cup(4, 8) \) (since the graph is below the x - axis from \( x=-9 \) to \( x = - 5 \) and from \( x = 4 \) to \( x = 8 \)).

Step3: Find the intersection of the two intervals

We need to find the intervals where both \( f'(x)\geq0 \) and \( f(x)\leq0 \) hold.

• For the interval where \( f'(x)\geq0 \) is \( (-9, - 7)\cup(4, 7) \) and \( f(x)\leq0 \) is \( (-9, - 5)\cup(4, 8) \).
• The intersection of \( (-9, - 7) \) and \( (-9, - 5) \) is \( (-9, - 7) \) (because \( - 7<-5 \)).
• The intersection of \( (4, 7) \) and \( (4, 8) \) is \( (4, 7) \).

Answer:

The intervals are \((-9, - 7)\) and \((4, 7)\)