Question

I've been working on this set of equations for about a month; interesting problem. It's an optimization problem. First, we have
$$\phi (x) = ((x-a)^2 + (y-b)^2 + (z-c)^2)^{1/2}$$

Answer 1

Now, the second derivative of this is:
$$\frac{\partial^2 \phi}{\partial x^2} = \frac{b^2 + c^2 - 2by + y^2 -2cz + z^2}{((a-x)^2 + (b-y)^2 + (c-z)^2)^{3/2}}$$

Answer 2

Now I have to solve such that:
$$\frac{\partial^2 \phi}{\partial x^2} < 0$$

Answer 3

In the actual problem, we're given \(x, y\) and \(z\), so treat them as real constants.

Answer 4

Oh yes, sorry. \(\phi\) is actually a function of x, y and z.

Answer 5

And we want to solve the inequality on a boundary \([\Delta x, \Delta y\, \Delta z)]\)

Answer 6

So, what now?

Answer 7

You said y, z are given, right?

Answer 8

x, y, and z are given. I need to solve for each variable.

Answer 9

What are the variables?

Answer 10

a, b and c.

Answer 11

Whether you consider a, b, c constants or x, y and z constants doesn't matter. Easily interchangeable. It may be more convenient to solve for x, y and z as variables.

Answer 12

You might want to look at this as a growing (or shrinking) sphere with the center changing where phi is the radius.
Or simply take partials in y, z to observe their minima and use that.

Answer 13

We're observing the maxima.

Answer 14

We're trying to find the maximal distance between two points on a boundary.

Answer 15

We might be able to set up a differential equation or something with the growing/shrinking sphere visualization.