Explore a function for extremums I have an exam tomorrow and have to know how to explore for extremums a function of 3 variables. For example: > Explore fo extremums the function: $$f(x,y,z)=x+ \frac{y^2}{4x} + \frac{z^2}{y} + \frac{2}{z}$$ Could someone explain me step by step what I have to do to solve the problem? Any help, would be greatly appreciated.

From $f_x(x,y,z)=0$, $f_y(x,y,z)=0$ and $f_z(x,y,z)=0$ we find $(\frac12, 1, 1)$ and $(-\frac12, -1, -1)$. Since for $x,y,z>0$ $$f(x,y,z)=x+\frac{y^2}{4x}+\frac{z^2}{2y}+\frac{z^2}{2y}+\frac{1}{2z}+\frac{1}{2z}+\frac{1}{2z}+\frac{1}{2z} \ge 8 \sqrt[8]{\frac{1}{2^8}}=4=f\left(\frac12, 1, 1\right)$$ the function has a local minimum at $(\frac12, 1, 1)$. Similarly $(-\frac12, -1, -1)$ is a point of local maximum.