Rolle and Lagrange

Question

Let $f$ be continuous and differentiable such that $f'+f''=0$. Show that there are constants $a$ and $b$ such that $f(x)=a\sin(x)+b\cos(x)$.

Any hints/ideas? Thanks.

Answer 1

This is false. Let $f(x)$ be identically equal to $17$.

Remark: If we replace the original equation by $f''(x)+f(x)=0$, then we can indeed conclude that $f(x)$ has shape $a\sin x+b\cos x$. To solve that problem, consider the function $f(x)-\left(f'(0)\sin x-f''(0)\cos x\right)$.