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# Minimum

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Among all pairs of numbers (x,y) such that 3x+y=15, find the minimum of x^2 + y^2.

Apr 28, 2022

#1
+124701
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3x + y  =  15

y = 15 - 3x

So

minimize   x^2 + (15 -3x)^2

minimize  x^2  + 9x^2  - 90x  +  225

minimize    10x^2  - 90x  +  225         take the derivative and  set to 0

20x  - 90  =  0

x = 90 / 20  = 9/2

x^2  =  81/4

And  y = 15 - 3(9/2)   =  15 - 27/2  =  3/2

y^2  =  9/4

So

min x^2 + y^2  =   (81/4) + (9/4)  =  90 / 4   =  45 / 2  =  22.5

Apr 28, 2022
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