SOLUTION: Suppose that A, B, and C are positive constants and that x+y= C. Show that the minimum value of {{{ Ax^2+By^2 }}} occurs when {{{ x= BC/(A+B) }}} and {{{ y=AC/(A+B) }}}

Question 626330: Suppose that A, B, and C are positive constants and that x+y= C. Show that the minimum value of occurs when and
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!
Suppose that A, B, and C are positive constants and that x + y = C. Show that the minimum value of Ax� + By� occurs when x = and y =

We know that if A > 0, the minimum value of y = Ax� + Bx + C 
occurs when x = 

To avoid conflict of letters we re-write that as

We know that if P > 0, the minimum value of y = Px� + Qx + R 
occurs when x = 

Since x + y = C, y = C - x, so

Minimum value of Ax� + By� = 

Minimum value of Ax� + B(C - x)� =

Minimum value of Ax� + B(C - x)(C - x) =

Minimum value of Ax� + B(C� - 2Cx + x�) =

Minimum value of Ax� + BC� - 2BCx + Bx� =

Minimum value of Ax� + Bx� - 2BCx + BC� =

Minimum value of (A + B)x� - 2BCx + BC� 

And by the rule above:

We know that if P > 0, the minimum value of y = Px� + Qx + R
 
occurs when x = 

Let P = (A + B),  Q = -2BC, and  R = BC�

Minimum value of (A + B)x� - 2BCx + BC� 

occurs when x =  =  = 

and since y = C - x, the value of y when x =  is

y = C -  =   -  =  

� -  =  -  =

 -  =  = .


Edwin

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