Exercise 4.3.7. [Used in Lemma 7.3.4 and Theorem 7.3.12.] Let [a, b] C R and [b, c] C R be non-degenerate closed bounded intervals, and let f: [a, b] - R and g: [b, c] - R be functions. Suppose that f(b) = g(b). Let h: [a, c] - R be defined by h(x ) = f ( x ) , if x E [a, b] g(x), if xe [b, c]. (1) Suppose that f and g are differentiable, and that f (b) = g'(b), where f (b) and g (b) are one-sided derivatives. Prove that h is differentiable.Exercise 4.3.6. [Used in Section 4.3.] Explain the flaw in the attempted proof of the Chain Rule (Theorem 4.3.3) that is given prior to the correct proof. Restate the Chain Rule with modified hypotheses that would make the attempted proof into a valid proof.Exercise 4.3.2. Let I f; R be an open interval, let r: E I, let it E N and let f1: . . . g f\" : I > R be functions. Suppose that - is differentiable at c for all i E {1, . . . ,n}. Prove that f1 f2 - - f\" is differentiable at c, and nd (and prove) a formula for (f1 f2 - - -fn)'(c) in terms off1(c),...,f,,(c) and ff(c,)...,'.;(c)