We have :

x^2+xy+y^2=3 and
y^2+yz+z^2=16
A=xy+yz+zx

Find the maximum value of

**A**.
Find x, y and z when A=max value.

(Remember the category)

I remember solving a simpler problem like this in highschool calculus.

Solve the first two equations leaving y as a parameter:

x = sqrt( 3 - 3(y^2)/4 ) - y/2

z = sqrt( 16 - 3(y^2)/4 ) - y/2

Substitute into A:

A = y*(sqrt (3 - 3*(y^2)/4) - y/2)

+ y*(sqrt (16 - 3*(y^2)/4) - y/2)

+ (sqrt (16 - 3*(y^2)/4) - y/2)*(sqrt (3 - 3*(y^2)/4) - y/2)

Now things get messy, find the first derivative of A. I had to use a symbolic calculator to make sure I got this right.

dA/dy = (-3*x/(4*sqrt(3-3*x^2/4))-1/2)*(sqrt(16-3*x^2/4)-x/2) +(sqrt(3-3*x^2/4)-x/2)*(-3*x/(4*sqrt(16-3*x^2/4))-1/2) +x*(-3*x/(4*sqrt(16-3*x^2/4))-1/2)+x*(-3*x/(4*sqrt(3-3*x^2/4))-1/2) +sqrt(16-3*x^2/4)+sqrt(3-3*x^2/4)-x

Numerically solving dA/dy = 0 yeilds A = 8.000000 and y = 0.718421 = sqrt(16/31) as the only maximum.