1. The problem statement, all variables and given/known data Let f(x)=log(x)+sin(x) on the positive real line. Use the mean value theorem to assure that for all M>0, there exists positive numbers a and b such that f(b)-f(a)/b-a=M 2. Relevant equations f'(x)=1/x+cos(x) 3. The attempt at a solution I know that as x→0, f'(x) gets arbitrarily large and as x→∞, f'(x) gets arbitrarily small, so for every M>0 there exists a and b such that f'(a)-M<0 and f'(b)-M>0. f'(x)-M would then have a root by the intermediate value theorem. From here, I don't know where to go. I know that as h→0, f(r+h)-f(r)/h→f'(r), where r is the root of f'(x)-M, so as h→0, the intervals (r,r+h) are such that f(b)-f(a)/b-a gets arbitrarily close to M by the Mean Value Theorem, but I don't know where to go. Any ideas?