minimize ( q = 3x^{2}+6y^{2} ), where ( x + y = 9 ).\n( x=)\n( y=)\n(simplify your answer. type an exact…

minimize ( q = 3x^{2}+6y^{2} ), where ( x + y = 9 ).\n( x=)\n( y=)\n(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers\nin the expression.)

minimize ( q = 3x^{2}+6y^{2} ), where ( x + y = 9 ).\n( x=)\n( y=)\n(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers\nin the expression.)

Answer

Explanation:

Step1: Express (y) in terms of (x)

From (x + y=9), we get (y = 9 - x).

Step2: Substitute (y = 9 - x) into (Q)

[ \begin{align*} Q&=3x^{2}+6(9 - x)^{2}\ &=3x^{2}+6(81-18x+x^{2})\ &=3x^{2}+486-108x + 6x^{2}\ &=9x^{2}-108x + 486 \end{align*} ]

Step3: Find the derivative of (Q) with respect to (x)

(Q^\prime(x)=18x-108)

Step4: Set the derivative equal to zero to find critical points

(18x-108 = 0) (18x=108) (x = 6)

Step5: Find (y)

Since (y=9 - x), when (x = 6), (y=9 - 6=3)

Answer:

(x = 6) (y = 3)