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) }}}

We know that if A > 0, the minimum value of y = Ax² + 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 = Px² + Qx + R 
occurs when x = 

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

Minimum value of Ax² + By² = 

Minimum value of Ax² + B(C - x)² =

Minimum value of Ax² + B(C - x)(C - x) =

Minimum value of Ax² + B(C² - 2Cx + x²) =

Minimum value of Ax² + BC² - 2BCx + Bx² =

Minimum value of Ax² + Bx² - 2BCx + BC² =

Minimum value of (A + B)x² - 2BCx + BC² 

And by the rule above:

We know that if P > 0, the minimum value of y = Px² + Qx + R
 
occurs when x = 

Let P = (A + B),  Q = -2BC, and  R = BC²

Minimum value of (A + B)x² - 2BCx + BC² 

occurs when x =  =  = 

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

y = C -  =   -  =  

· -  =  -  =

 -  =  = .


Edwin