Inequality with Differentiation

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For the inequality xy ≤ xp/p + yq/q, where p > 1 and q = p/(p-1), the goal is to demonstrate this relationship for positive x and y. The approach involves using differentiation to analyze the function f(x,y) = x^p/p + y^q/q - xy. Finding the minimum value of this function will help establish the conditions for equality. The discussion emphasizes the need for a solid understanding of differentiation techniques to tackle the problem effectively. The equality holds when x and y are proportional, specifically when x^p/p = y^q/q.
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Homework Statement


Let p > 1, and put q = p/(p-1), so 1/p + 1/q = 1. Show that for any x > 0, y > 0, we have

xy <= xp/p + yq/q, and find the case where equality holds.

Homework Equations



The Attempt at a Solution


This is in the differentiation chapter of my analysis book (Browder), so I'm going to go out on a limb here and assume that some aspect of differentiation comes into play here. Unfortunately, I don't really know how to start. Could someone get me started here? Thanks!
 
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Maybe try to find the minimum possible value of f(x,y)=x^p/p + y^q/q -xy
 

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