i am just out of ideas here... that is the question, its from a test in my college
let $ f(x), g(x)$ be differentiable functions. for each $x$ it is true that $ f'(x)g(x) \neq g'(x)f(x)$. We also know that $ f(a)=f(b)=0$
prove that exists a $c, a<c<b$ so that $g(c)=0$
Ive tried to build some functions to use the Intermediate value theorem or Lagrange theorem. but no success...