Respuesta :
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.
⇒ [tex]g_x=18x[/tex]
Now, we find the partial derivative of function g(x, y) with respect to x.
⇒ [tex]g_y=12y[/tex]
To find the critical point:
consider [tex]g_x=0[/tex] and [tex]g_y=0[/tex]
⇒ 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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