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PROVE THAT X^2+2XY+3Y^2-6X-2Y CANNOT BE LESS THAN -11. X,Y BELONG TO ALL REAL NOS.....THANKS FOR HELP!

PROVE THAT X^2+2XY+3Y^2-6X-2Y CANNOT BE LESS THAN -11. X,Y BELONG TO ALL REAL NOS.....THANKS FOR HELP!

Grade:11

1 Answers

mycroft holmes
272 Points
13 years ago

Equivalently we have to prove that 2x2+4xy+6y2-12x-4y+22>=0

 

From Cauchy Schwarz (CS) Inequality, 4 [(x+2y)2+(y-1)2+(y-1)2+(x-6)2] >= (x+2y+1-y+1-y+6-x)2 = 64

 

Hence, (x+2y)2+(y-1)2+(y-1)2+(x-6)2 >= 16

 

or 2x2+4xy+6y2-12x-4y+22>=0

 

Since the CS equality condition cannot be met here, we have 2x2+4xy+6y2-12x-4y+22>0

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