Hi there, if I'm not wrong you should do the following:
first - find the derivative of f(x)
second - use the Mean Value Theorem f(c) = (f(1) - f(0))/(1-0),
see here the proof Mean value theorem - Wikipedia, the free encyclopedia
The integral of your function from 0 to 1 is approximately 0.55, so it would seem that you could set 0 < f(0) < 0.45 and f'(x) slightly greater than g(x) to get a function which stays between 0 and 1, but the derivative is never equal to g(x).
I took my own advice and surfed the web for the intermediate value theorem for derivatives. The proof in wikapedia, Fermat's theorem (stationary points) - Wikipedia, the free encyclopedia, has an error or at least an incomplete argument. PlanetMath has the same proof. So here's a complete proof of the theorem.