If a function \( f \) is increasing on the intervals \( [-3,3] \) and \( [9,17] \) and decreasing on the interval \( [3,9] \), determine the interval(s) on which \( g(x)=-2 f(2 x-5) \) is increasing. The graph \( g(x)=-2 f(2 x-5) \) is increasing on interval (Type your answer in interval notation.)

We are given f is increasing on [-3,3] and [9,17] and decreasing on [3,9]. The function g is defined by   g(x) = -2 f(2x - 5). Step 1. Rewrite g(x) in terms of u:   Let u = 2x - 5. Then g(x) = -2 f(u). Step 2. Compute g′(x) using the chain rule:   g′(x) = -2 f′(u) · (d/dx)(2x - 5) = -2 f′(u) · 2 = -4 f′(2x - 5). Step 3. Find when g is increasing:   g increases when g′(x) > 0.   So, -4 f′(2x - 5) > 0 ⟹ f′(2x - 5) < 0. Since f′(u) < 0 when f is decreasing, we require that u = 2x - 5 is in an interval where f is decreasing. We are told that f is decreasing on [3,9]. Thus, we set:   3 ≤ 2x - 5 ≤ 9. Step 4. Solve the inequality for x:   Add 5 to all sides: 3 + 5 ≤ 2x ≤ 9 + 5 ⟹ 8 ≤ 2x ≤ 14.   Divide all sides by 2: 4 ≤ x ≤ 7. Thus, g(x) is increasing for x in the interval [4,7]. Answer in interval notation: [4,7].