Putnam 1976 A6

A-6. Suppose $f(x)$ is a twice continuously differentiable real valued function defined for all real numbers $x$ and satisfying $|f(x)| \leqq 1$ for all $x$ and $(f(0))^{2}+\left(f^{\prime}(0)\right)^{2}=4$. Prove that there exists a real number $x_{0}$ such that $f\left(x_{0}\right)+f^{\prime \prime}\left(x_{0}\right)=0$.