Question

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

Answer 1

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.