Question

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

Explanation:

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

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

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

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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:

\(x = 6\)
\(y = 3\)