6 In this question you may take for granted the usual

Last updated: 6/8/2023

6 In this question you may take for granted the usual properties of the exponential function and d d r a Let a 0 and consider g x a Show that Here I am identifying log ee VER d d9 x log a a In 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 R c Show that there exists some x 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