Questions: problem #4: The graph of f is given to the right. Which of the below graphs is a graph of its derivative f'?

Solution Steps

The given graph of f(x) appears to be a cubic function with a local maximum roughly around x=-1 and a local minimum around x=1.

Since the function f(x) has a local maximum around x=-1, its derivative, f'(x), must have a zero and be changing from positive to negative at x=-1. f(x) also has a local minimum around x=1, so f'(x) must have a zero and be changing from negative to positive at x=1. Lastly, since f(x) appears to be a cubic, its derivative f'(x) should be a parabola (a quadratic function).

The only graph that satisfies the conditions described above is (G). It is a parabola, it crosses the x-axis (i.e., equals zero) at approximately x=-1 and x=1, and it is negative between x=-1 and x=1 and positive elsewhere.

Final Answer: The final answer is $\boxed{G}$