Question 67977: Could someone please please helpme this:
1. Let a,b,c be the three distinct positive real numbers.
Thank you very for your great help!!!!!!!
a) Prove that a^3+b^3>a^2xb+b^2xa.
b) Write down , without proving them, similar inequalities for (a^3+c^3) and (b^3+c^3).
c) Hence, or otherwise, prove that 3(a^3+b^3+c^3)>(a^2+b^2+c^2)(a+b+c)
2.State why (a-b)^2(a^2+b^2) is non-negative, for all a,b belong to R. Deduce that a^4+b^4is greater or equal to 2(a^3xb-a^2xb^2+axb^3).
Thanks a million!!! I am waiting for a solution for ages. Please help me , i am desperate...
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Could someone please please helpme this:
1. Let a,b,c be the three distinct positive real numbers.
Thank you very for your great help!!!!!!!
a) Prove that a^3+b^3>a^2xb+b^2xa.................1
TST
A^3+B^3-A^2B-B^2A>0
TST
A^2(A-B)-B^2(A-B)>0
TST
(A^2-B^2)(A-B)>0
TST
(A+B)(A-B)(A-B)>0
TST
(A+B)(A-B)^2>0
IN LHS (A-B)^2 IS ALWAYS POSITIVE AS IT IS A PERFECT SQUARE AND A AND B ARE DISTINCT
A+B IS ALSO POSITIVE AS A AND B ARE POSITIVE
HENCE THEIR PRODUCT IS ALWAYS +VE...PROVED
b) Write down , without proving them, similar inequalities for (a^3+c^3) and (b^3+c^3).
A^3+C^3>A^2C+C^2A.................2
B^3+C^3>B^2C+C^2B................3
ADDING 1,2,3
2(A^3+B^3+C^3)>
c) Hence, or otherwise, prove that 3(a^3+b^3+c^3)>(a^2+b^2+c^2)(a+b+c)
ADDING 1,2,3
2(A^3+B^3+C^3)>A^2B+B^2A+B^2C+C^2B+C^2A+A^2C
ADDING A^3+B^3+C^3 TO BOTH SIDES
3(A^3+B^3+C^3)>A^3+B^3+C^3+A^2B+B^2A+B^2C+C^2B+C^2A+A^2C=(A^2+B^2+C^2)(A+B+C)
THE LAST ONE IS A STANDARD FORMULA..IF YOU WANT ITS PROOF PLEASE COME BACK..YOU CAN MULTIPLY AND CHECK FOR YOUR SELF.
2.State why (a-b)^2(a^2+b^2) is non-negative, for all a,b belong to R.
(A-B)^2 BEING A PERFECT SQUARE>=0
A^2+B^2=SUM OF PERFECT SQUARES WHICH IS >=0
HENCE THEIR PRODUCT IS>=0
Deduce that a^4+b^4is greater or equal to 2(a^3xb-a^2xb^2+axb^3).
(A-B)^2(A^2+B^2)=(A^2+B^2-2AB)(A^2+B^2)
=A^4+A^2B^2+B^2A^2+B^4-2A^3B-2AB^3>=0.........AS PROVED
A^4+B^4>=2(A^3B-A^2B^2+B^3A).....PROVED
Thanks a million!!! I am waiting for a solution for ages. Please help me , i am desperate...
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