4. Recall Taylor's Theorem in one variable, which states the following. Taylor's Theorem in one variable. Let / C R be an open interval and let x* E I. Let f : I - R be n + 1 times continuously differentiable. Then, for any r E I f (x) = f(a*) + f'(x*) (x - x*)+ f"(z*) (x - x*) 2+... + f (7 ) ( 20 * ) 2! n! (x - x*)" + Rn(x) with Rn(2) = f(n+1) (2) (x - 2*)n+1 (n + 1)! where z is a point between x and x*. (a) Consider the case n = 1 and show that if f'(a*) = 0 and furthermore f"(x) > 0 for all x E I, then x* is a strict global minimizer of f on the open interval I. (b) Apply the above result on the example of f(x) = ex