Yahoo Answers: Answers and Comments for More Inequality Proofs...? [Mathematics]
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From Creeping Sandman
enUS
Thu, 14 May 2009 05:56:19 +0000
3
Yahoo Answers: Answers and Comments for More Inequality Proofs...? [Mathematics]
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https://answers.yahoo.com/question/index?qid=20090514055619AAr5kSt
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From Nino_ops: Im not sure abt what r u asking in the first Q...
https://answers.yahoo.com/question/index?qid=20090514055619AAr5kSt
https://answers.yahoo.com/question/index?qid=20090514055619AAr5kSt
Thu, 14 May 2009 14:28:19 +0000
Im not sure abt what r u asking in the first Q
Why its given on a b anc and asking for x y and z ? ?
are ther more details like a+bc > 0 ? ?
or a b and c are sides of a triangel ? ?
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Use the AG GM inequality on
1). a^2 and b^4 [ a^2 ≥ 0 , b^4 ≥ 0]
(a^2 + b^4)/2 ≥ sqrt(a^2 b^4)
(a^2 + b^4)/2 ≥ ab^2
2). b^4 and c^6 [ b^4 ≥ 0 , c^6 ≥ 0]
(b^4 + c^6)/2 ≥ sqrt(b^4 c^6)
(b^4 + c^6)/2 ≥ b^2 c^3
3). c^6 and d^8 [ c^6 0 , d^8 ≥ 0]
(c^6 + d^8)/2 ≥ sqrt(c^6 d^8)
(c^6 + d^8(/2 ≥ c^3 d^4
4). d^8 and a^2
(a^2 + d^8)/2 ≥ sqrt(a^2 d^78)
(a^2 + d^8)/2 ≥ ad^4
Now add 1 2 3 4 together
(a^2 + b^4 + b^4 + c^6 + c^6 + d^8 + d^8 + a^2)/2 ≥ ab^2 + b^2 c^3 + c^3 d^4 + ad^4
(a^2 + b^4 + c^6 + d^8) ≥ ab^2 + b^2 c^3 + c^3 d^4 + ad^4
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Let x y and z be any real positive non zero numbers.
F(x , y) = x^3 + y^3 – x^2y – y^2x
F(x , y) = x^2 (x – y)  y^2(x – y)
F(x , y) = (x – y)(x^3 – y^3)
F(x , y) = (x – y)(x – y)(x^2 + y^2 + xy)
F(x , y) = (x – y)^2 (x^2 + y^2 + xy)
(x^2 + y^2 + xy) is always positive
(x – y)^2 is always positive but it is 0 when x = y
Therefore;
F(x , y) ≥ 0
(x – y)^2 (x^2 + y^2 + xy) ≥ 0
x^3 + y^3 – x^2y – y^2x ≥ 0
Since we prooved this for any x y z (x,y,x positive nonzero numbers)
we can use it on a b and c
a^3 + b^3  a^2b  b^2a ≥ 0
a^3 + c^3  a^2c c^2a ≥ 0
c^3 + b^3  c^2b  b^2c ≥ 0
adding them together;
2 (a^3 + b^3 + c^3)  a^2b  b^2a  a^2c c^2a  c^2b  b^2c ≥ 0
2(a^3 + b^3 + c^3) ≥ a^2b +b^2a + a^2c + c^2a + c^2b + b^2c