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 x can have ?
(Corrected per SilverKnight's comment (#8)):
Substituting z=5xy into xy+yz+xz=3 gives
xy+y(5xy)+x(5xy)=3
which can be simplified to
y²+(5x)y+5xx²3=0
Considered as an equation to be solved for y, but without actually solving, we get a discriminant of 13+10x3x² This must be positive in order for there to be a real solution for y.
The bounds of where this is positive are those two points where it is zero. So, setting that discriminant to zero, we get
x=(10±√256)/6 = (10±16)/6
or x = 13/3 or 1
Between these two values of x, the discriminant is positive, so y has a real value (as well as z).
So in answer to the question, the largest value that x can have is 13/3.
Edited on November 18, 2003, 9:12 am

Posted by Charlie
on 20031117 15:31:51 