minimize ( q = 2x^{2}+4y^{2} ), where ( x + y = 6 )\n( x=)\n( y=)\n(simplify your answer. type an exact…

minimize ( q = 2x^{2}+4y^{2} ), where ( x + y = 6 )\n( x=)\n( y=)\n(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression )

minimize ( q = 2x^{2}+4y^{2} ), where ( x + y = 6 )\n( x=)\n( y=)\n(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression )

Answer

Explanation:

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

From (x + y=6), we get (y = 6 - x).

Step2: Substitute (y = 6 - x) into (Q)

[ \begin{align*} Q&=2x^{2}+4(6 - x)^{2}\ &=2x^{2}+4(36-12x+x^{2})\ &=2x^{2}+144-48x + 4x^{2}\ &=6x^{2}-48x + 144 \end{align*} ]

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

(Q^\prime(x)=12x-48)

Step4: Set the derivative equal to zero and solve for (x)

(12x-48 = 0) (12x=48) (x = 4)

Step5: Find (y)

Since (y=6 - x), when (x = 4), (y=6-4=2)

Answer:

(x = 4) (y = 2)