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Three numbers (Posted on 2003-11-17) Difficulty: 3 of 5
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 ?

No Solution Yet Submitted by Ravi Raja    
Rating: 3.7500 (4 votes)

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Solution Another Solution Comment 21 of 21 |
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.

  Posted by Brian Smith on 2016-02-21 15:49:21
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