Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 3

Table of contents
- Solution 3

Solution

The denominator

D = x^{2} + x y + y^{2} = (x + y /2)^{2} + 3 y^{2} /4

is positive away from the origin. The partial derivatives are

f_{x} = \frac{y ( y ^{2} - x ^{2} )}{D ^{2}}, f_{y} = \frac{x ( x ^{2} - y ^{2} )}{D ^{2}} .

They both vanish exactly on

y = x and y = - x, (x, y) \neq = (0, 0) .

These are lines of stationary points, with values

1/3

and

- 1

, respectively.

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