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Use lagrange multiplier techniques to find the local extreme values of f(x, y) = x2 − y2 − 2 subject to the constraint x2 + y2 = 16

User Durrell
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Given
f(x,\ y)=x^2-y^2-2 subject to the constraint
x^2+y^2=16

Let
g(x,\ y)=x^2+y^2.

The gradient vectors of
f and
g are:


\\abla f(x,\ y)=\langle2x,-2y\rangle and
\\abla g(x,\ y)=\langle2x,2y\rangle

By Lagrange's theorem, there is a number
\lambda, such that


\langle2x,-2y\rangle=\lambda\langle2x,2y\rangle=\langle2\lambda x,2\lambda y\rangle


\lambda=\pm1

It can be seen that
f(x,\ y)=x^2-y^2-2 has local extreme values at the given region.
User Hamstar
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