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Let x and y be real numbers such that 2(x^2 + y^2) = x + y + 1. Find the maximum value of x - y.

 Apr 14, 2022
 #1
avatar+122 
-4

Chris got u he composing an answer rn

 Apr 14, 2022
 #2
avatar+122390 
+2

I believe this is correct.....

 

2 (x^2 + y^2)  = x + y + 1

 

2x^2 - x + 2y^2 - y   = 1       complete the square on x ,y

 

2(x^2 - (1/2)x + 1/16)  + 2(y^2 - (1/2)y + 1/16)   = 1  + 1/4

 

2 ( x - 1/4)^2 + 2 (y - 1/4)^2  =  5/4          divide through by 2

 

(x - 1/4)^2  + (y -1/4)^2 =  5/8

 

We have a circle centered at   ( 1/4 , 1/4)   with a radius of  sqrt (5/8)

 

x - y    will  be maxed when

 

x  = rcos(-45°)      and y = r cos (-45°)

 

x = sqrt (5/8)(sqrt (1/2)  = sqrt (5) / sqrt (16)  = sqrt 5/4

y = sqrt (5/8) (-sqrt (1/2)) =  -sqrt (5) / sqrt (16) =  -sqrt (5)/4

 

So

 

x - y  max   =   (sqrt (5)  - - sqrt (5) ] / 4  =   2sqrt (5) / 4  = sqrt (5) / 2

 

cool cool cool

 Apr 14, 2022
edited by CPhill  Apr 14, 2022

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