Find the absolute maximum and minimum values of the function, subject to the given constraints. g(x,y)=9x2 6y2; −1≤x≤1 and −1≤y≤

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2 years ago

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1 answer:

blondinia [14] · Answer 1 · 2023-12-26T02:10:36+03:00

For function g(x, y) = 9x² + 6y²,

the absolute minimum is 15 and the absolute maximum is 303

For given question,

We have been given a function g(x, y) = 9x² + 6y² subject to the constraint −1≤x≤1 and −1≤y≤7

We need to find the absolute maximum and minimum values of the function.

First we find the partial derivative of function g(x, y) with respect to x.

⇒ $g_x=18x$

Now, we find the partial derivative of function g(x, y) with respect to x.

⇒ $g_y=12y$

To find the critical point:

consider $g_x=0$ and $g_y=0$

⇒ 18x = 0 and 12y = 0

⇒ x = 0 and y = 0

This means, the critical point of function is (0, 0)

We have been given constraints −1 ≤ x ≤ 1 and −1 ≤ y ≤7

Consider g(-1, -1)

⇒ g(-1, -1) = 9(-1)² + 6(-1)²

⇒ g(-1, -1) = 9 + 6

⇒ g(-1, -1) = 15

And g(1, 7)

⇒ g(1, 7) = 9(1)² + 6(7)²

⇒ g(1, 7) = 9 + 294

⇒ g(1, 7) = 303

Therefore, for function g(x, y) = 9x² + 6y²,

the absolute minimum is 15 and the absolute maximum is 303

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