How can I prove that for any real ( x , y ) root this...

Brock Byrd

Brock Byrd

Answered

2022-06-29

How can I prove that for any real ( x , y ) root this system of equations
x 4 + y 2 = ( a + 1 a ) 3
x 4 y 2 = ( a 1 a ) 3
the inequation x 2 + | y | 4 is always true?
Also, when will x 2 + | y | = 4 be true for this system?

Answer & Explanation

Keegan Barry

Keegan Barry

Expert

2022-06-30Added 18 answers

Hint: x 4 = ( x 4 + y 2 ) + ( x 4 y 2 ) 2 and y 2 = ( x 4 + y 2 ) ( x 4 y 2 ) 2
So x 4 = a 3 + 3 a 1 a 2 = a 3 + 3 1 a and y 2 = 3 a 2 1 a + 1 a 3 = 3 a + 1 a 3 (and presumably a > 0 else we'd have 1 0 or x 4 < 0)
So x 2 + | y | = a 3 + 3 1 a + 3 a + 1 a 3 so by AM-GM
x 2 + | y | 2 ( a 3 + 3 1 a ) ( 3 a + 1 a 3 ) 4 = 2 3 a 4 + 9 + 1 + 3 a 4 4
And applying the AM-GM result that for x > 0 that
x + 1 x 2 x 1 x = 2 we have
x 2 + | y | 2 3 a 4 + 9 + 1 + 3 a 4 4 = 2 10 + 3 ( a 4 + 1 a 4 ) 4 2 10 + 3 2 4 = 2 16 4 = 2 2 = 4
racodelitusmn

racodelitusmn

Expert

2022-07-01Added 5 answers

Hint:
Add and subtract the equations and divide by 2 to get
x 4 = a 3 + 3 a and y 2 = 1 a 3 + 3 a

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