Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

Quartics

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Three Ways

Stage: 5 Challenge Level:

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

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

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 easily verifies that this indeed gives a maximum so the maximum value is $1/4$.

Method 3
By the AM-GM inequality, we have

$(xy)^{(1/2)} \leq (x + y)/2$

so $xy \leq (-1/2)^2 = 1/4.$