A popular technique arising in methods for minimizing functions in several variables involves a weak line search, where an approximate minimum x∗ is found for a function in one variable, f (x), for which the values of f (0), f (0), and f (1) are given. The function f (x) is defined for all nonnegative x, has a continuous second derivative, and satisfies f (0) ∗ as the minimum of the interpolant.
(a) Find x∗ for the values f (0) = 1, f'(0) = -1, f (1) = 2.
(b) Show that the quadratic interpolant has a unique minimum satisfying 0 ∗