Answer :
The maximum value of f(x, y) = xy on the ellipse 82 + y² = 2 is 2 and the minimum value is -2.
To find the maximum and minimum values of f(x, y) on the given ellipse, we need to use the method of Lagrange multipliers. The objective function is f(x, y) = xy and the constraint function is g(x, y) = 82 + y² - 2 = 0.
Using Lagrange multipliers, we get the following system of equations:
fx = λgx
fy = λgy
g(x, y) = 0
where fx and fy are the partial derivatives of f with respect to x and y, and gx and gy are the partial derivatives of g with respect to x and y. λ is the Lagrange multiplier.
Solving these equations, we get:
y = ± √2
x = ± √2
Substituting these values in f(x, y) = xy, we get:
f(√2, √2) = 2
f(-√2, -√2) = 2
f(-√2, √2) = -2
f(√2, -√2) = -2
Therefore, the maximum value of f(x, y) on the given ellipse is 2 and the minimum value is -2.
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