# Taylor expansion, thermodynamics

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## Homework Statement

I'm asked to show that the relation ##S(U+\Delta U, V+ \Delta V , n )+S(U-\Delta U, V- \Delta V , n ) \leq 2 S(U,V,n)## implies that ##\frac{\partial ^2 S}{\partial U ^2 } \frac{\partial ^2 S}{\partial V ^2}- \left ( \frac{\partial ^2 S }{\partial U \partial V} \right ) \geq 0##.

## Homework Equations

Taylor expansion of a function of several variables: https://en.wikipedia.org/wiki/Taylor_series#Taylor_series_in_several_variables

## The Attempt at a Solution

The book gives some help, it says to first make a Taylor expansion of the left side to the second order in ##\Delta U## and ##\Delta V##. So I should reach the equation ##S_{UU}(\Delta U)^2+2S_{UV}\Delta U \Delta V + S_{VV} (\Delta V)^2 \leq 0##, then I'd have to rewrite this expression and I should reach the answer.
However I'm having a problem at understanding what the book means by "in ##\Delta U## and ##\Delta V##". Does it means I should expand S and U around the point ##(\Delta U , \Delta V, n)##?
Or...

Edit: I've expanded S around the point (U,V,n) which seemed to make sense to me, but then I don't reach what I should. Instead I reach ##-S+\Delta U S_U + \Delta V S_V + \frac{1}{2} \left [ (\Delta U)^2 S_{UU} + 2 \Delta U \Delta V S_{UV} + (\Delta V )^2 S_{VV} \right ] \leq 0## where the functions are evaluated in ##(U,V,n)##.

Edit2: NEVERMIND!!!! I forgot a term to make the Taylor expansion. I didn't do it yet but at first glance this is the right thing to do, i.e. making the expansion around (U,V,n)!
Edit3: problem solved, I've reached the final result!!!

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