1. ## Optimization

When working an optimization problem, when should I consider the domain of the primary equation?

Also, when it is determined that the interval of which the variable is an element is open, do I consider the endpoints? For example. Let's say that $f$ is continuous on the open interval $(a,b)$ and has a relative maximum at $x=c$. But at the point $x=d$, with $a, $f(d)>f(c)$. This is strange because, when I differentiate with respect to the variable and find all values for which $f'(x)=0$, all I get is the relative extrema, And I can't consider the endpoints because they are not in the domain. What do I do?

2. Originally Posted by VonNemo19
When working an optimization problem, when should I consider the domain of the primary equation?

Also, when it is determined that the interval of which the variable is an element is open, do I consider the endpoints? For example. Let's say that $f$ is continuous on the open interval $(a,b)$ and has a relative maximum at $x=c$. But at the point $x=d$, with $a, $f(d)>f(c)$. This is strange because, when I differentiate with respect to the variable and find all values for which $f'(x)=0$, all I get is the relative extrema, And I can't consider the endpoints because they are not in the domain. What do I do?
If the domain is a closed interval you use the 'closed interval method', where you essentially list the y-values of all critical numbers and the y-values of the endpoints, then pick out the least or the greatest.

I am not sure I completely understand the second part of your question. Remember that critical numbers are x values for which f'(x) is zero or undefined.

My two cents.

3. Originally Posted by apcalculus
I am not sure I completely understand the second part of your question.
What I'm saying is that an open interval can contain a relative max, but not an absolute max. Think of the extreme value theorem, if continuity goes, then so does the theorem.

So, again, when optimizing, if there be a relative maximum on $(a,b)$ then there is some point, call it c, such that $f'(c)=0$. this is considered a relative maximum at $x=c$. But, consider what happens if there is another number (or an infinite set of numbers), call it $d$ in (a,b) for which $f(d)>f(c)$, you can see that the Optimization method has failed at locating the absolute maximum on $(a,b)$. The problem is that the endpoints are non-inclusive.

Do you see what I mean now. I tried to express myself as best I could.

4. Yes. You are correct. Absolute (or global) extrema need not exist when the function is only defined in an open interval. The closed interval with continuity guarantees the existence of both types of global extrema. (Extreme Value Theorem)

f(x) = -x^2 + 1 on (-1, 0) union (0, 1)
f(0) = 1/2

... is an example of a function with no types of global extrema.

Good luck!