Problem 7. Suppose f is continuous on [0, 1] and differentiable on (0, 1). ( a) Prove that if {In} is a Cauchy sequence in [0, 1] then {f (x'n) } is...

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Problem 7. Suppose f is continuous on [0, 1] and differentiable on (0, 1).( a) Prove that if {In} is a Cauchy sequence in [0, 1] then {f (x'n) } is a Cauchy sequence.(b) If f(0) = 0 and f(1) = 2, prove that there is a point x E (0, 1) with3f (x)2 - 4f (x) = -1.(c) If f(0) = 0 and f(1) = ;, prove that there is a point c E (0, 1) with6f (c)f' (c) - 4f'(c) = -5