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Calculus — II, , Dr. Vijay S ;, Assistant Professor of Mathematics,, , Government Science College,, Hassan-573 201

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Statement: If f:[a, b] — Ris such that, , 1. f(x) Continuous on [a, b], 2. f(x) is differentiable on (a, b), 3. f(a) = f(b), , then there exists c € (a,b) such that f’(c) = 0.

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Let f:[a, b] — R be such that, 1. f(x) Continuous on [a, b], , 2. f(x) is differentiable on (a, b), , 3. f(a) = f(b), , Since the function f(x) is continuous in [a,b], it will be, bounded in [a, b] and it attains its infimum=m(say) and it attains its, supremum=M(Say) on [a, b|, , -. there exists a, 8 € [a, b] such that m = f(a) and M = f(B)

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Case i) If m = M, then f is a constant on [a, b]., , => f'(x) =0 V x € (a,b), .. there exist c € (a,b) such that f' (c) =0, , and hence theorem is proved.

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Case ii) If m # M, But f(a) = f(b), , , , Then either m # f(a) andm # f(b) |, |, , , , or, , , , M + f(a) and M + f(b), , , , Suppose m # f(a) and m # f(b), => f(a) # f(a) and f(a) # f(b), => azx#aand ax#b_ Buta é [a,b], , Hence a € (a,b) , But f is differentiable on (a, b), , “ F'(@) exists, , .. f is differentiable at a