6. In this question, you may take for granted the usual properties of the exponential function and d ex = e, d.x Vx R. (a) Let a > 0 and consider g(x) = a". Show that d dx9(x) = == (log a)a". Here, I am identifying log = ln. Hint: Write a = eloga and use the chain rule. (b) Consider the function h(x)=1+x+2. Show that h'(x) 1. Hint: log 2 > 0 and 20 for all x E R. (c) Show that there exists some x E [-2,0] such that h(x) = 0. Hint: Intermediate Value Theorem. (d) Prove that the root of h in the previous part is the unique root of h on R. Hint: Part b) says that h is strictly increasing, see Lemma 4.2.1 of the notes.