High School

Let [tex]f(x, y) = xy - 42[/tex] on [tex]D = \{(x, y) : x^2 + y = 1, y \neq 0\}[/tex].

[tex]M[/tex] = absolute maximum of [tex]f(x, y)[/tex]

[tex]m[/tex] = absolute minimum of [tex]f(x, y)[/tex]

Find [tex]M + 2m[/tex].

A. 8
B. 12
C. 16

Answer :

Final answer:

The absolute maximum and minimum of the function f(x, y) = xy? - 42 on the given domain D = {(1, y) : x² + y = 1, Y0} are both -42. Therefore, M + 2m = -126.

Explanation:

To find the absolute maximum and minimum of the function f(x, y) = xy? - 42 on the given domain D = {(1, y) : x² + y = 1, Y0}, we need to analyze the critical points and endpoints of the function.

First, let's find the critical points by taking the partial derivatives of f(x, y) with respect to x and y:

∂f/∂x = y - 42

∂f/∂y = x - 42

Setting these partial derivatives equal to zero, we get:

y - 42 = 0

x - 42 = 0

Solving these equations, we find that the critical point is (42, 42).

Next, let's analyze the endpoints of the domain D. The domain is defined as {(1, y) : x² + y = 1, Y0}. Since x is fixed at 1, we only need to consider the values of y that satisfy the equation x² + y = 1:

1² + y = 1

y = 0

Therefore, the endpoint of the domain is (1, 0).

Now, we evaluate the function at the critical point and endpoint:

f(42, 42) = 42 * 42? - 42 = 42 * 0 - 42 = -42

f(1, 0) = 1 * 0? - 42 = -42

Since both values are the same, -42, it means that -42 is both the absolute maximum (M) and absolute minimum (m) of the function.

Finally, we calculate M + 2m:

M + 2m = -42 + 2(-42) = -42 - 84 = -126

Learn more about finding the maximum and minimum values of a function here:

https://brainly.com/question/17811972

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