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. (15 marks) Let a > U be a positive real number and f: IR. > R be a differentiable function with f(0) 2.1' ' (a) Prove that for any a: E (0, a), there exists 0 E (0, :12) such that (c)_ M) 1 . (b) Suppose f\"(t) > 0 for any real numberm t. Prove that for any a: E (0, a), we have at)