Find the roots r1, r2, r3, r4 of the equation 4x4 - ax3 + bx2 - cx + 5 = 0, given that they are positive reals satisfying r1/2 + r2/4 + r3/5 + r4/8 = 1.
Solution
We have r1r2r3r4 = 5/4 and hence (r1/2) (r2/4) (r3/5) (r4/8) = 1/44. But AM/GM gives that (r1/2) (r2/4) (r3/5) (r4/8) ≤ ( (r1/2 + r2/4 + r3/5 + r4/8)/4 )4 = 1/44 with equality iff r1/2 = r2/4 = r3/5 = r4/8. Hence we must have r1 = 1/2, r2 = 1, r3 = 5/4, r4 = 2.
© John Scholes
jscholes@kalva.demon.co.uk
1 July 2002