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Find the minimum value of the expression $x^2+y^2+2x-4y+8+10x-12y$ for real $x$ and $y$.

 Feb 12, 2024
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x^2  + y^2  + 2x - 4y + 8 + 10x  -12y

 

x^2 + 12x + y^2 - 16y  +  8         

 

Take the derivative with respect to x and  set to 0

 

2x + 12  =  0    →   x + 6 = 0  →  x  = -6

 

Take the derivative with respect  to y and  set to  0

 

2y  - 16  =  0   →  y  -  8  = 0  →   y =  8

 

The minimum occurs at   (-6,8)

 

And the minimum is

 

(-6)^2 + 12 (-6) + (8)^2  - 16(8) + 8 =    -92

 

cool cool cool

 Feb 12, 2024

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