Although my previous post on this topic didn't cause a discussion I'll try to continue. If no one will be interested in problems from this post too, I'll switch to another theme.
Today I would like to present several inequalities directly involving integrals.
Problem 4. If is a continuous function, prove the following inequality
Solution. To start our proof we'll consider the following function We know that , so to prove our inequality we simply need to prove taht is a decreasing fucntion. However, we know that range of is , so , subsequently by Theorem 1. . So which means that is decreasing and . Finaly we arrvie at the desired result
Now try your own techniques to solve the following problems.
Problem 5. For continuous, differentiable function , there exists such that . Prove that
Hint. For this problem you might consider the Mean Value Theorem
Problem 6. Let be a continuous function satisfying following property Prove that
Hint. Try to search for a linear fucntion with the same proprties, also here .