How do we determine whether we are getting the minimum value or the maximum value of a function using lagrange...

Question

today I've solved a simple question like what is the maximum value of $abc$ if $a+b+c=10$ and $a,b,c$ are positive real numbers?

The arithmetic geometric mean solution is trivial, and one can easily decide whether you get a minimum or a maximum.

But when I used lagrange like $$f(a,b,c,\lambda)=abc+\lambda(a+b+c-10)$$ and took the derivatives accordingly, I got $a=b=c$, that's okay. But without knowledge of the relation between arithmetic and geometric means how do I know when I am finding maximum or minimum? Sometimes we get absolute value but sometimes we do not...

What are your suggestions? Thank you in advance...

Answer 1

Any time you find extreme points using derivatives, what you find are the critical points: that is, the candidates to be extreme (i.e., if there is a maximum or minimum, it will be in the list). To the list of critical points you found, you also need to add the boundary of your region.

In this case, since you require $a\geq0$, $b\geq0$, and $c\geq0$, the boundaries $a=0$, $b=0$, $c=0$ should be considered. Each of these give you the minimum $abc=0$, so the extreme you found is a maximum.