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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:
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