11No, for the second one its 36 only.
You've done a minor mistake.
It should be (ab+bc+cd+ad+ac+bd)2 ≥ 36abcd
Given that the equation x^4+px^3+qx^2+rx+s=0 has four real, positive roots, prove that-
(a)pr-16s\geq 0
(a)q^2-36s\geq 0
Is there any proof without using Cauchy-Schwarz?
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3 Answers
21ah
1)pr≥ 16s
iff (a+b+c+d)(abc+ bcd+ cda + adb)≥ 16(abcd)
by AM-GM
a+b+c+d ≥ 4(abcd)(1/4)
aind abc+ bcd+ cda + adb ≥ 4(abcd)(3/4)
multiply them..so its true..qed
2) q2≥ 36s
iff (ab+bc+ cd+ ad)2 ≥ 36abcd
By AM-GM ab+ bc+ cd+ ad ≥ 4(abcd)(1/2)
square it down.. so you probably meant 16 and not 36
or i have missed something?
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