$f(0)^2+f'(0)^2=4$ then $f(x_0)+f''(x_0)=0$

Question

If $f(x)$ is a real valued $C^2$ function on the real line such that its value is between $-1$ and $1$. Then, if $f(0)^2+f'(0)^2=4$, I have to show that there exists $x_0$ on the real line such that $f(x_0)+f''(x_0)=0$. I think I have to use a very ingenious maximum-minimum theorem for $g(x)=f(x)^2+f'(x)^2$ but I am just stuck.... I cannot find a way to solve this problem....Could anyone please help me? It's so frustrating...