5. Suppose that f is differentiable on R.

(a) If f'(x) = 0 for all x E R, prove that f(x) = f(O) for all x E R.

(b) If f(O) = 1 and 1f'(x)1 :::; 1 for all x E R, prove that If(x)1 :::; Ixl + 1 for all x E R.

(c) If f'(x) ::::: 0 for all x E R, prove that a < b implies that f

(a) :::; f(b).