Given f(0)=f’(0)=0, f’(x)>0, x>0
Show: f(x)>0, x>0 using std calculus (no abstract symbolism please).
Question: Is continuous on and differentiable on ? (This is true for ).
If so, then we can apply the Mean Value Theorem. Let . By assumption, is continuous on and differentiable on , so by the Mean Value Theorem, there exists such that:
Solving for , we have . Since , we are given that . The product of two positive real numbers is positive, so .