Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.


b. Does either g or h have a local extreme value at x = 2? Explain.

Verified step by step guidance

1. Since f has a local extreme value at x = 2, we know that f'(2) = 0. This is a key point to analyze the functions g and h.
2. To determine if g has a local extreme value at x = 2, we need to compute g'(x) using the product rule: g'(x) = f(x) + x f'(x).
3. Evaluate g'(2) by substituting x = 2 into the derivative: g'(2) = f(2) + 2 f'(2). Since f(2) = 0 and f'(2) = 0, we find that g'(2) = 0.
4. Next, we check if h has a local extreme value at x = 2 by computing h'(x): h'(x) = f(x) + x f'(x) + 1.
5. Evaluate h'(2) by substituting x = 2 into the derivative: h'(2) = f(2) + 2 f'(2) + 1. Since f(2) = 0 and f'(2) = 0, we find that h'(2) = 1.