Maximum point of a function

Question

I am looking to find the local maximum value of the function

$$f(x,y,z) = 9+ \frac{(x-y)^2}{xy} + \frac{(y-z)^2}{yz}+ \frac{(z-x)^2}{zx}$$

at a point within the cube $[a,b] \times [a,b] \times [a,b],$ where $0<a<b$. I took the partial derivative of f with respect to x, y, and z. I set those to zero. I used the second-derivative test for multi-variable functions.

The second-derivative test requires the computation of a 3 by 3 matrix. (Because the function is a three-variable function)

$D= \det \pmatrix{f_{xx} & f_{xy} &f_{xz} \\ f_{yx} & f_{yy} &f_{yz} \\ f_{zx} & f_{zy} &f_{zz}}$

I found that the first dervatives equal to zero at $x=y=z$. On the other hand, Wolfram Alpha confirms that the max for the $[1,3] \times [1,3] \times [1,3]$ is for $y=z=\frac{x}{3}$ and $x=\frac{y}{3}=\frac{z}{3}$. I got confused

Answer 1

$$\frac{\partial^2f}{\partial x^2}=\frac{2(y+z)}{x^3}>0.$$ Similarly, $\frac{\partial^2f}{\partial y^2}>0$ and $\frac{\partial^2f}{\partial z^2}>0.$

Thus, $f$ is a convex function of $x$, of $y$ and of $z$, which says: $$\max_{(x,y,z)\in[a,b]^3}f=\max_{\{x,y,z\}\subset\{a,b\}}f=f(a,a,b)=$$ $$=9+\frac{(a-a)^2}{a\cdot a}+\frac{(a-b)^2}{ab}+\frac{(a-b)^2}{ab}=9+\frac{2(a-b)^2}{ab}.$$ A local maximum does not exist.

A local minimum is equal to $9$.

Answer 2

At $x=y=z$ you have a minimum. You can see this by plugging in these into the equation for $f$ and you get $9$. For any positive values, as long as at least two are distinct, you have $f(x,y,z)>9$. What you want to do is look at the points on the boundary of the cube. Due to symmetry, just look at the $x=a$ and $x=b$ planes.

Answer 3

We compute \begin{align} f_{xx} &= f_x\left(\frac1y-\frac y{x^2}\right) = \frac{2y}{x^3},\\\\ f_{xy} &= f_y \frac{(y+z) \left(x^2-y z\right)}{x^2 y z} = \frac{-(x^2+y^2)}{x^2y^2}\\ f_{xz} &= f_x \frac{(x+y) \left(z^2-x y\right)}{x y z^2} = \frac{-(x^2-z^2)}{x^2 z^2}\\ f_{yz} &= f_y \frac{(x+y) \left(z^2-x y\right)}{x y z^2} = \frac{-(y^2+z^2)}{y^2 z^2}. \end{align} Since the mixed partials are continuous, by Clairaut's theorem we need only compute \begin{align} f_{yy} &= f_y\frac{(x+z) \left(z^2-x z\right)}{x y^2 z} = \frac{2 (x+z)}{y^3}\\ f_{zz} &= f_z\frac{(x+y) \left(z^-2x y\right)}{x y z^2} = \frac{2 (x+y)}{z^3}. \end{align} Hence the Hessian matrix is $$ H = \begin{pmatrix} \frac{2y}{x^3} & \frac{-(x^2+y^2)}{x^2y^2} & \frac{-(x^2-z^2)}{x^2 z^2}\\ \frac{-(x^2+y^2)}{x^2y^2} & \frac{2 (x+z)}{y^3} & \frac{-(y^2+z^2)}{y^2 z^2}\\ \frac{-(x^2-z^2)}{x^2 z^2} & \frac{-(y^2+z^2)}{y^2 z^2} & \frac{2 (x+y)}{z^3}. \end{pmatrix} $$ We have $f_{xx} = \frac{2y}{x^3}>0$ on $[a,b]^3$ and $$ \begin{vmatrix} f_{xx} & f_{xy}\\\ f_{xy}& f_{xz} \end{vmatrix} = \begin{vmatrix} \frac{2y}{x^3} &\frac{-(x^2+y^2)}{x^2y^2}\\\ \frac{-(x^2+y^2)}{x^2y^2} & \frac{-(x^2-z^2)}{x^2 z^2} \end{vmatrix} = \frac{\left(x^2+y^2\right) \left(-x \left(x^2+y^2\right)-\frac{2 y^5}{z^2}\right)}{x^5 y^4}, $$ which takes negative values. So $F$ admits no global minimum. As shown in @Michael Rozenberg's answer, $9$ is a local minimum.