Problem 6. (log-sum-exp/softmax function.) Consider the function (z) = log (?-1 ei) (also known as the softmax function in ML). Oftenm the convexity of this function is proven by deriving the Hessian. In this problem, you will show that f is convex using another method. a. Show that, for any given s > 0, we have b. Show that log s = 1 + min (se v) . V n f(z) = 1+ min ( v ezi+v o). - V c. Prove convexity of f based on the above result. Hint: Use the convexity-preserving property called "partial minimization, this is Module 4 lecture 2 slides, p. 16 (which we'll cover on Monday).