Extension: Integrating Factors for Differential Equations

(HMMT February 2002, Calculus Round, #9) Suppose $f$ is a real differentiable function such that $f(x) + f'(x) \le 1$ for all $x$, and $f(0) = 0$. What is the largest possible value of $f(1)$?

(This is clearly a differential inequality, not a differential equation, but it uses almost exactly the same principle!)

When you think you have the answer, check the answer and solution to see if you are correct. Feel free to look at the hint if you are completely stuck.