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SOLVED:Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Also, suppose that f(a)=f(b) and that c is a real number in the interval such that f^'(c)=0 . Find an interval for the function g over which Rolle's Theorem can be applied, and find the corresponding critical number of g(k is a constant). (a) g(x)=f(x) k (b) g(x)=f(x-k) (c) g(x)=f(k x)
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(c) Let f be a real-valued continuously differentiable function mapping the closed interval [a, b] to itself. Suppose there is a constant 0 < M < 1 such that |f'(x)| < M for each x ∈ [a, b]. Show that f has a unique fixed point.Let f be a real-valued continuously differentiable function defined on the closed interval [a, b]. Suppose that f(a) and f(b) have opposite signs. Suppose in addition that 0 < M1 < |f'(x)| < M2 for each x ∈ [a, b]. Show that f has a unique root in [a, b]. (Hint: For a
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