Given P = ( x + 3 y − 5 ) 2 − 6 x y + 2 6 . Find the minimum value of P
Bonus: Which value of x , y can achieve this?
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There are no constraints, so partial differentiation is the easiest (but probably not the most elegant) way to solve this. We have
P x = 2 ( x + 3 y − 5 ) − 6 y = 2 ( x − 5 ) P y = 6 ( x + 3 y − 5 ) − 6 x = 6 ( 3 y − 5 )
Setting both of these equal to zero we get x = 5 , y = 3 5 and substituting in, P = 1 (it's easy to see from the form of P that this must be a minimum).
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P = ( x + 3 y − 5 ) 2 − 6 x y + 2 6 = x 2 + 9 y 2 + 6 x y − 1 0 x − 3 0 y + 2 5 − 6 x y + 2 6 = x 2 + 9 y 2 − 1 0 x − 3 0 y + 5 1 = ( x 2 − 1 0 x + 2 5 ) + ( 9 y 2 − 3 0 y + 2 5 ) + 1 = ( x − 5 ) 2 + ( 3 y − 5 ) 2 + 1
Therefore,
min ( P ) = min ( ( x − 5 ) 2 + ( 3 y − 5 ) 2 + 1 ) = min ( ( x − 5 ) 2 ) + min ( ( 3 y − 5 ) 2 ) + 1 = 0 + 0 + 1 = 1 when x = 5 , y = 3 5