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Putnam 1976 A6

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April 18, 2022
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Problem Statement

Suppose f(x)f(x) is a twice continuously differentiable real valued function defined for all real numbers xx and satisfying ∣f(x)∣≤1|f(x)| \leq 1 for all x and (f(0))2+(f′(0))2=4.(f(0))^2+(f'(0))^2=4. Prove that there exists a real number x0x_0 such that f(x0)+f′′(x0)=0.f(x_0)+f''(x_0)=0.
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