Perfect Harmony

Let $\mathcal F$ be the collection of all nonnegative decreasing functions $\psi$ defined on $[a,b]$ such that $\int_a^b\psi(x)dx=1$ and $a\psi(a)=b\psi(b)(0<a<b).$ For any $\psi,\phi\in\mathcal F,$ consider the integral \[I(\psi,\phi)=\int_a^b\max[\psi (x),\phi (x)]dx.\] As $\psi$ and $\phi$ range over $\mathcal F,$ find the largest possible value of $I(\psi,\phi).$

Try similar problems Perfect Harmony and II.

#Calculus

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Vijay Simha - 1 year, 11 months ago

Could you please tell me how to get this result in detail?

Brian Lie - 1 year, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$