Find the absolute maximum and minimum of the function f(x, y)=x^2−y^2f(x, y)=x^2−y^2 subject to the constraint x^2+y^2=121x^2+y^2=121.

The absolute maximum of f(x,y) is 121. The absolute minimum of f(x,y) is -121

Step-by-step explanation:

The given function f(x,y) can be seen as a quadratic form:

The constraint can be seen as:

Using the Min-max theorem with Rayleigh–Ritz quotient, we can easly obtain the absolute maximum and minimum of a quadratic form:

Therefore:

So the problem is reduced to obtain the maximum and minimum eigenvalues of the matrix A.

This eigenvalues can be obtained directly (diagonal matrix), where and . Therefore:

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step-by-step explanation: