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**Apache** An individuals utility depends upon the consumption of goods x and y,

$\displaystyle U=2x$$\displaystyle ^{2}+4xy-4y^{2}+64x+32y-14$

I have to find the utility maximisation of x and y,

$\displaystyle \frac{\partial u}{\partial x}=-4x+4y+64$ (1)

$\displaystyle \frac{\partial u}{\partial y}=4x-8y+32$ (2)

Multiply (1) by 2 = $\displaystyle 128+8y-8x$ (3)

$\displaystyle 32-8y+4x$ (4)

(3)-(4), after rearranging and substitution,

x = 8, y = 8

To check these are the maximum points,

$\displaystyle \frac{\partial ^2y}{\partial x^2}=-4$

$\displaystyle \frac{\partial ^2x}{\partial y^2}=-8$

Find cross partial derivitives,

$\displaystyle Uxx = -4$

$\displaystyle Uyy = - 8$

$\displaystyle Uxy = 4$

$\displaystyle UxxUyy-Uxy^{2}$

= 16 > 0

3 conditions meet = maximum

I then have to use the total differential of the function to find an expression for the slope of any curve.

Slope of curve = $\displaystyle \frac{dy}{dx}$

$\displaystyle du=\frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy=0$ (**)

This is where i become a little stuck as to what I should do next.