Question #2357

Find the maximum and minimum values of the function f(x,y) = x2 ‐2y subject to the constraint
that x2 + y2 = 9 and please show calculations

Expert's answer

We should use Lagrange multipliers and Extreme value theorem. Let's construct the system of equations:

2x = 2λ x;

-2 = 2λ y;

x^{2} + y^{2} = 9.

Hence: λ = 1, y = -1, x^{2} = 8.

The function has one extreme value:

f(x,y) = 8 + 2 = 10.

f(√5, 2) = 5 - 2 = 3.

Thus f(x,y) = 10 is the maximum of the given function subject to constraint that x^{2} + y^{2} = 9.

2x = 2λ x;

-2 = 2λ y;

x

Hence: λ = 1, y = -1, x

The function has one extreme value:

f(x,y) = 8 + 2 = 10.

f(√5, 2) = 5 - 2 = 3.

Thus f(x,y) = 10 is the maximum of the given function subject to constraint that x

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