I'm being asked to show this inequality on the way towards proving Young's inequality. I see in other posts there is usually a given about about the sum of the RHS, but this is the prompt I was given.
Let $p > 1$ and $q > 1$. Show that $\frac{x^p}{p} + \frac{y^q}{q} \geq \frac{1}{p} + \frac{1}{q}$ for all $x > 0$ and $y > 0$ with $xy = 1$.
I've tried numerous algebraic tricks and it always seems to be insufficient information. Any insights appreciated.
$\frac{x^p}{p} + \frac{y^q}{q} \geq \frac{1}{p} + \frac{1}{q} $
$\begin{array}\\ \frac{x^p}{p} + \frac{y^q}{q} &=\frac{x^p}{p} + \frac{x^{-q}}{q}\\ &=f(x)\\ \end{array} $
If $x = y = 1$, $f(x) = \frac1{p}+\frac1{q}$.
Therefore we can assume that $x, y \ne 1$.
Choose $x$ so that $x \gt 1$; if not, swap $x$ and $y$.
$f'(x) =x^{p-1}- x^{-q+1} =x^{p-1}(1- x^{-q+1-(p-1)}) =x^{p-1}(1- x^{-q-p+2}) =0 $ for $x=1$.
Since $x > 1$, and $p+q > 2$, then $x^{-q-p+2} \lt 1$ so that $f'(x) > 0$.
Therefore $f(x)$ is increasing for $x > 1$ so it is a minimum at $x = 1$ where $f(1) =\frac1{p}+\frac1{q} $.