Prove the following inequalities using the mean value theorem:
- , for .
- Proof. Define and . Then and are continuous and differentiable everywhere so we may apply the mean value theorem. We obtain
The final step follows since for all
- Proof. Let , , then and . So, by the mean-value theorem we have there exists a such that,
But, since is an increasing function on the positive real axis, and we have we know
Further, since and is positive we can multiply all of the terms in the equality by without reversing inequalities to obtain,
Therefore, substituting from above we obtain the requested inequality: