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 f(x) = x2 - 8x.

Here's the solution for question 1.b): 1.b) 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 f(x) = x^2 - 8x. Step 1: Find the first derivative of f(x). f(x) = x^2 - 8x f'(x) = (d)/(dx)(x^2 - 8x) = 2x - 8 Step 2: Find the critical numbers by setting f'(x) = 0. 2x - 8 = 0 2x = 8 x = 4 The critical number is x=4. Step 3: Determine the intervals of increasing and decreasing using a sign chart for f'(x). We test values in the intervals (-, 4) and (4, ). • For x < 4, let's pick x=0: f'(0) = 2(0) - 8 = -8. Since f'(0) < 0, f(x) is decreasing on (-, 4). • For x > 4, let's pick x=5: f'(5) = 2(5) - 8 = 10 - 8 = 2. Since f'(5) > 0, f(x) is increasing on (4, ). Step 4: Identify relative extrema using the First Derivative Test. Since f'(x) changes from negative to positive at x=4, there is a relative minimum at x=4. The value of the function at x=4 is: f(4) = (4)^2 - 8(4) = 16 - 32 = -16 Summary: • Critical number: x=4 • Increasing interval: (4, ) • Decreasing interval: (-, 4) • Relative extrema: Relative minimum at (4, -16) That's 2 down. 3 left today — send the next one.