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)

(In reply to

Most of a solution by Brian Smith)

There are two maximum points on A. They are given by y=4/sqrt 31 and y=-4/sqrt 31. These are the only maxima on the curve A.

x=7/sqrt 31 y=4/sqrt 31 z=20/sqrt 31 A=8

x=-7/sqrt 31 y=-4/sqrt 31 z=-20/sqrt 31 A=8

This has been confirmed by solving dA/dy for all cases.