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.


a. Evaluate g(2), h(2), g'(2), and h'(2).

Verified step by step guidance

1. First, evaluate g(2) by substituting x = 2 into the function g(x) = xf(x) + 1. This gives g(2) = 2f(2) + 1.
2. Since f(2) = 0, simplify g(2) to find its value.
3. Next, evaluate h(2) by substituting x = 2 into the function h(x) = xf(x) + x + 1. This gives h(2) = 2f(2) + 2 + 1.
4. Again, use the fact that f(2) = 0 to simplify h(2) and find its value.
5. To find g'(2) and h'(2), apply the product rule and the sum rule of differentiation to g(x) and h(x) respectively, then substitute x = 2 into the derivatives.