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The conjugate exponent of $1<p<\infty$ is $q=p/(p-1)$, so $1/p+1/q=1$. The endpoint conventions pair $1$ with $\infty$ and $\infty$ with $1$. Examples are $(p,q)=(2,2)$, $(3,3/2)$ and $(4,4/3)$. The exponents themselves need not add to one; their reciprocals do.

Use a normalization proof of Young's inequality. For arbitrary $x,y\ge 0$, putIf $A=0$, both coordinates are zero and the desired inequality is equality. If $A>0$, defineThen $X^p/p+Y^q/q=1$. Part (c) gives $XY\le 1$, and the conjugate exponent identity gives $XY=xy/A$. Therefore Young's inequality for products isThe unique equality case in part (c) says $X=Y=1$. Translated back, and including the zero case, this gives the sharp conditionConversely, if both powers equal $r\ge 0$, then $xy=r^{1/p+1/q}=r$ and the right side is $r(1/p+1/q)=r$, directly verifying equality.