Prove that the equation $\cos x - x -1/2 = 0$ has a unique real solution.
My solution: I was starting with a function $F(x) = \cos x - 1/2$ and the interval $[0,\pi/4]$ and trying to show that the fixed point theorem is applicable here. If I can find this solution to a few decimal places then I have shown that there is unique real solution. However, I am having some trouble applying the Fixed Point Theorem to $F(x)$. Any help would be greatly appreciated.