Boundary considerations in extremum problems

[silver-rose]

Homework Statement

We are given a word problem and asked find maxima/minima (ie a simple example would be to find the least amount surface area required to build a box of a given volume).

Is it necessary to explicitly show that the relative interior max/min, calculated by setting the gradient to 0, is also the absolute extremum by evaluating possible extrema on the boundaries of the domain of the function, even when the physical considerations of the problem render it blatantly obvious that considering the boundaries will not yield a reasonable answer?

For example, for the aforementioned box problem,

x = length of box
y = width of box
z = height of box

The Domain of the box is

D --> { x,y,z : x [tex]\geq0[/tex], y[tex]\geq0[/tex], z[tex]\geq0[/tex] }

Must we explicitly evaluate boundary situations for when x=0 , y=0, z=0 ? (In this case, we see immediately that the volume will be 0) What about for the cases where x is large?

Do I also need to take the limit of x, y, and z to infinity?

 

[EnumaElish]

Is it necessary to explicitly show that the relative interior max/min, calculated by setting the gradient to 0, is also the absolute extremum by evaluating possible extrema on the boundaries of the domain of the function, even when the physical considerations of the problem render it blatantly obvious that considering the boundaries will not yield a reasonable answer?

You should solve the boundary conditions, however simple. In a case where it is blatantly obvious that the extrema are not on the boundary, you may include a simple proof that, for example, the function is uniformly zero on the boundary, but positive on an interior point.

Do I also need to take the limit of x, y, and z to infinity?

This is a good question. Suppose you are asked to maximize f(x) = x² over x > 0. Obviously, there is no maximum. Then, you have a choice between stating that f has no maximum, and stating that f is maximized when x becomes infinite.