optimize the objective function (3x^{2}+y^{2}-2xy) subject to the constraint (6 = x + y)

Answer

Explanation:

Step1: Substitute (y = 6 - x) into the objective function

Substitute (y=6 - x) into (z=3x^{2}+y^{2}-2xy). [ \begin{align*} z&=3x^{2}+(6 - x)^{2}-2x(6 - x)\ &=3x^{2}+36-12x+x^{2}-12x + 2x^{2}\ &=(3x^{2}+x^{2}+2x^{2})+(-12x-12x)+36\ &=6x^{2}-24x + 36 \end{align*} ]

Step2: Find the derivative of (z) with respect to (x)

Differentiate (z = 6x^{2}-24x + 36) using the power rule ((x^{n})^\prime=nx^{n - 1}). (z^\prime=\frac{d}{dx}(6x^{2}-24x + 36)=12x-24)

Step3: Set the derivative equal to zero and solve for (x)

Set (z^\prime = 0), so (12x-24 = 0). Add (24) to both sides: (12x=24). Divide both sides by (12): (x = 2).

Step4: Find the second - derivative of (z)

Differentiate (z^\prime=12x - 24) with respect to (x). (z^{\prime\prime}=\frac{d}{dx}(12x-24)=12>0). Since (z^{\prime\prime}>0), the function (z) has a minimum at (x = 2).

Step5: Find the value of (y)

Substitute (x = 2) into (y=6 - x), we get (y=6-2 = 4).

Step6: Find the minimum value of the objective function

Substitute (x = 2) and (y = 4) into (z=3x^{2}+y^{2}-2xy). [ \begin{align*} z&=3\times2^{2}+4^{2}-2\times2\times4\ &=12 + 16-16\ &=12 \end{align*} ]

Answer:

The minimum value of the function (3x^{2}+y^{2}-2xy) subject to the constraint (x + y=6) is (12) when (x = 2) and (y = 4).