Is my claim true about the sign of the function?

Question

I have a continuous function $f(x)$ for $0<x<a$ where $f(0)=0$. I want to know whether this function is positive or negative. From numerical data, I know that $f'(x)$ is negative, but it is not easy to prove it, but I can prove that $f'(x)$ can not be zero (this gives a contradiction). Then, does this means that the function is monotonic (either increasing or decreasing)? If so, computing $f(a)$ will clear if the function is positive or negative. Is this claim true?