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 ?
Let x*y*z=p. Then consider the cubic function f(c) = c^3 - 5c^2 + 3c - p. Its roots will be x, y, and z. The effect of changing the value of p is a vertical translation of f(c).
A cubic function d=f(c) has three real roots as long as its two relative extrema are not both on the same side of c-axis. The two boundary cases are when one of the extrema is tangent to the c-axis. These two cases correspond to when the smallest root is at its minimum or when the largest root is at its maximum; the root of the derivative will be a double root of f(c).
f'(c) = 3c^2 - 10c + 3 = 0 yields c = 3 or 1/3. Then (x,y,z) = (-1,3,3) or (13/1,1/3,1/3). The largest x can be is 13/3, and the smallest it can be is -1.