a,b,x,y, are negative numbers a^5 + b^5<= 1 x^5 + y^5 <= 1 prove that a^2x^3 + b^2y^3<= 1

Tammy Todd 2021-02-25 Answered
a,b,x,y, are negative numbers
a5+b51
x5+y51
prove that a2x3+b2y31
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Expert Answer

Roosevelt Houghton
Answered 2021-02-26 Author has 106 answers

The numbers a2andb2 will always be positive as these are squared numbers.
The numbers x3andy3 will always be negative numbers as it is the cube of negative numbers and the cube of negative numbers is always negative numbers.
x3<0 andy3<0 if x,y<0
Therefore,
a2x3<0andb2y3<0 if x,y<0 .
The sum of negative numbers is also negative . Therefore,
a2x3+b2y3<0 (1)
Now if a=x, b=y. Then,
a2x3+b2y3=a2(a3)+b2(b3)
=a5+b5
1
if a=x,b=y,a2x3+b2y31 (2)
Now combine the results of equation (1) and (2).
Hence, a2x3+b2y31 has been proved.

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