Minimize Q = x2 + 2ysquared subject to x+y = 5. Answer to 2 decimal places.?

Question

Minimize Q = x2 + 2ysquared subject to x+y = 5. Answer to 2 decimal places.

yes, you are right this first one is what i mean... poor typing on my part

thank you very much

Answer 1

Do you mean:
Q = x^2 + 2y^2
Q = x^2 + (2y)^2
Q =(x^2 + 2y)^2
?

I'll pick the first one, but the principle will be the same.

Rewrite with y = 5-x (from the contraint).
Q = x^2 + 2(5-x)^2
Q = x^2 + 2(25 - 10x + x^2)
Q = x^2 +50 -20 x + 2x^2
Q = 3x^2 -20 x + 50

Differenciate:

dQ = (6 x -20) dx
dQ/dx = 6x - 20

A minimum (or maximum) for Q relative to x can only occur when dQ/dx = 0

0 = 6x - 20

solve for x

set y = 5 - x

check that it is a minimum (not a maximum) by changing the values of x and 5-x a little on either side.