Find the critical numbers of f, if any, find the open intervals on which the function is increasing/decreasing, apply the First Derivative Test to identify all relative extrema for g(x) = x2 - 2x - 8.

Welcome back — been a while! Let's pick up where you left off. Here's the solution for question 1.a): 1.a) Find the critical numbers of f, if any, find the open intervals on which the function is increasing/decreasing, apply the First Derivative Test to identify all relative extrema for g(x) = x^2 - 2x - 8. Step 1: Find the first derivative of g(x). g(x) = x^2 - 2x - 8 g'(x) = (d)/(dx)(x^2 - 2x - 8) = 2x - 2 Step 2: Find the critical numbers by setting g'(x) = 0. 2x - 2 = 0 2x = 2 x = 1 The critical number is x=1. Step 3: Determine the intervals of increasing and decreasing using a sign chart for g'(x). We test values in the intervals (-, 1) and (1, ). • For x < 1, let's pick x=0: g'(0) = 2(0) - 2 = -2. Since g'(0) < 0, g(x) is decreasing on (-, 1). • For x > 1, let's pick x=2: g'(2) = 2(2) - 2 = 4 - 2 = 2. Since g'(2) > 0, g(x) is increasing on (1, ). Step 4: Identify relative extrema using the First Derivative Test. Since g'(x) changes from negative to positive at x=1, there is a relative minimum at x=1. The value of the function at x=1 is: g(1) = (1)^2 - 2(1) - 8 = 1 - 2 - 8 = -9 Summary: • Critical number: x=1 • Increasing interval: (1, ) • Decreasing interval: (-, 1) • Relative extrema: Relative minimum at (1, -9) Send me the next one 📸