1. The problem statement, all variables and given/known data Given 2 functions f(x) and g(x) that are differentiable everywhere on R and f′(x) = g(x) and g′(x) = −f(x). Prove that 1. Between any two consecutive zeros of f(x)=0 there is exactly one zero of g(x)=0, 2. Between any two consecutive zeros of g(x)=0 there is exactly one zero of f(x)=0. 2. Relevant equations 3. The attempt at a solution I guess the first question has something to do with Rolle's Theorem but the theorem only states that there exists a zero of f'(x)=0 between 2 zeros of f(x), without mentioning about the uniqueness of that zero. Also I have trouble tackling the second question. Any help is appreciated, thanks!