# Relative maximum and minimum of function of three variables

I know that how to find relative maximum and minimum of function of two variables.

How can I determine function when $f(x,y,z)=x^2+y^2-z^2$ has relative maximum or relative minimum?

Please give me hint. In general when does $f(x,y,z)$ have relative minimum or relative maximum? Thanks in advance.

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What do you do in two dimensions? –  Siminore Nov 3 '12 at 9:10
@Siminore In two dimensions:Suppose f (x, y) is a function and (a, b) is a point where and Let 1. If D>0 and , then $f_{xx}>0$ has a relative minimum at (a, b). 2. If D>0 and , then $f_{yy}<0$ has a relative maximum at (a, b). 3. If D<0 , then f has neither a local maximum or local minimum. The function f has a saddle point at (a, b). 4. If D=0 , the test fails. –  Trivedi Nov 3 '12 at 9:13
What? What is "If and,"? –  Siminore Nov 3 '12 at 9:16
This function is radially symmetric, for each $x$, $y$ such that $x^2+y^2=r^2$ (i.e., $x=r\cos\varphi$, $y=r\sin\varphi$) it has the same value. So if it helps you to visualize the problem, you could think about the function $g(r,z)=r^2-z^2$ first. –  Martin Sleziak Nov 3 '12 at 9:17
just edited....where $D=f_{xx}f_{yy}-(f_{xy})^2$ –  Trivedi Nov 3 '12 at 9:18

Have a look at Hessian Matrix and see under "Second derivative test".

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