Integral Equations and the Limits of Athletic Performance

Problem: Given $f:[0,1] \to \mathbb{R}$ is integrable over $[0,1]$, and that The way to the solution here is not trivial. I started by always recognizing that, with integral inequalities, it never hurts to start with the integral of something squared is greater than or equal to zero. Well, given the integral constraints above, we can guess what we need to do. Expand the integrals and use the info provided to get that We then maximize the expression on the right. We may maximize by taking derivatives of $g(a,b)$ wrt $a$ and $b$ and setting them equal to zero. It is then easy to verify that $g(6,-2)=4$. The assertion is proven. for any $a,b \in \mathbb{R}$. (). Continue reading.

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