Question 3 (a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x) #0 for any re(a, b). Determine whether there exists k € (a, b) such that f(k)-f(a) f'(k) g(b)-g(k) g'(k)* (Hint: consider the function h: [a, b] → R defined by = h(z) = f(r)g(r) – f(a)g(2) – g(b)f(z), 2 € a,b], and compute h'.) (b) If f(x) = z² and g(x) 23 with a [-1, 1], find the c € (-1,1) by using Cauchy's Mean Value Theorem. (c) For the functions in part (b), determine whether there exists c € (-1, 1) such that f(1)-f(-1) f'(c) g(1) g(-1) g'(c) =

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A: Solution :-

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