Question

Define g(x) = f(x) + tan^-1 (2x) on [−1, √3/2 ]. Suppose that both f'' and g'' are continuous for all x-values on [−1, √3/2 ]. Suppose that the only local extrema that f has on the interval [−1, √3/2 ] is a local minimum at x = 1/2 .

(a) Determine the open intervals of increasing and decreasing for g on the interval [1/2 , √3/2] .

(b) Suppose f(1/2) = 0 and f(√3/2) = 2. Find the absolute extrema for g on [1/2 , √3/2] . Justify your answer.

(c) Suppose f(0) = −5, f'(0) = −2, g''(0) = 7. What can we say about the point (0, g(0)) on the graph of g? Be as specific as possible. Justify your answer. Hint: Find g' (0)

Answer 1