Here are some solutions from Koopa,Boston College, USA. Vassil, Lawnswood Sixth Form, Leeds sent in similar methods. Can you find a co-ordinate geometry (i.e. graphical) method or yet another different method?

Method 1 We have x + y = -1. So, to maximise xy, I need to maximize

-x(x + 1) = -[(x + 1/2)2 - 1/4] = -(x + 1/2)2 + 1/4,

so, xy is maximized at x = -1/2 and the maximum value is 1/4.

Method 2 Let f(x) = -x(x + 1), then by differentiation f¢(x) = -2x - 1 and to find a maximum or minimum f¢(x) = 0 gives x = -1/2. The second derivative test easi ly verifies that this indeed gives a maximum so the maximum value is 1/4.

Method 3 By the AM-GM inequality, we have

(xy)(1/2) £ (x + y)/2

so

xy £ (-1/2)2 = 1/4.