Functions from derivatives Use the derivative f' to determine the x-coordinates of the local maxima and minima of f, and the intervals on which f is increasing or decreasing. Sketch a possible graph of f (f is not unique).


f'(x) = 10 sin 2x on [-2π, 2π]

Verified step by step guidance

1. Identify the critical points by setting the derivative f'(x) = 10 sin(2x) equal to zero and solving for x. This will help find where the function f has potential local maxima and minima.
2. Determine the intervals where f'(x) is positive or negative by analyzing the sign of 10 sin(2x). This will indicate where the function f is increasing or decreasing.
3. Use the critical points found in step 1 to test the intervals identified in step 2. Choose test points in each interval to determine the behavior of f'(x).
4. Classify the critical points as local maxima or minima by applying the First Derivative Test: if f' changes from positive to negative at a critical point, it is a local maximum; if it changes from negative to positive, it is a local minimum.
5. Sketch a possible graph of f based on the information gathered about the critical points and the intervals of increase and decrease, ensuring to reflect the local maxima and minima identified.