Given [tex]f(x,\ y)=x^2-y^2-2[/tex] subject to the constraint [tex]x^2+y^2=16[/tex]
Let [tex]g(x,\ y)=x^2+y^2[/tex].
The gradient vectors of [tex]f[/tex] and [tex]g[/tex] are:
[tex]\nabla f(x,\ y)=\langle2x,-2y\rangle[/tex] and [tex]\nabla g(x,\ y)=\langle2x,2y\rangle[/tex]
By Lagrange's theorem, there is a number [tex]\lambda[/tex], such that
[tex]\langle2x,-2y\rangle=\lambda\langle2x,2y\rangle=\langle2\lambda x,2\lambda y\rangle[/tex]
[tex]\lambda=\pm1[/tex]
It can be seen that [tex]f(x,\ y)=x^2-y^2-2[/tex] has local extreme values at the given region.
