Search for local extrema

Question

Investigate for local extrema (don't forget to check for maximum/minimum) the following function: $F(x,y,z) = \frac{xy + xz^2 + y^2z}{xyz} + x +1$ To specify all the points of local maxima and minima.Response to justify.

I used the standard method for solving this problem: I found partial derivatives and critical points, then I used the Hesse matrix and the Sylvester criterion to find out which of them were the maxima and minima. There are points : $M_1 = (-1;1-1)$ $M_2 = (1;1;1)$. Where M2 is local minimum, M1 is local maximum. But I am not sure that this is all, and I do not know what additional analysis is needed to show that there are no more points.

Answer 1

On $D= \mathbb R^3 \setminus \{(0,0,0)\}$, $f$ is smooth. So the local extrema are given by critical points and Hesse matrix on the open subset $D$. And you said that you found those. Good!

Now $$\lim\limits_{t \to 0^+} f(t,t,t) = \infty \text{ and } \lim\limits_{t \to 0^-} f(t,t,t) = -\infty.$$

So $f$ is not bounded.

Answer 2

Since $F$ has partial derivatives for all $(x,y,z)$ with $xyz\ne 0$, the local extremum points satisfy $F_x=0, F_y=0, F_z=0$. Solving this system, we obtain that the possible extremum points are $M_1=(-1,1,-1)$ and $M_2=(1,1,1)$. There are no other candidates because these are all the solutions to the system, and the partial derivatives exist for all $(x,y,z)$ in the domain of $F$.

So, the next step is to check whether $M_1$ and $M_2$ are extremum points using the Hessian. Based on what you say, you've already done this part.