Given that $x + y = -1$ find the largest value of $xy$
(a) by co-ordinate geometry
(b) by calculus
(c) by algebra.
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
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 easily verifies that this indeed gives a
maximum so the maximum value is $1/4$.