I have an optimization question that was mostly answered, can someone answer the rest of it really quickly?

[When doing optimization, how would you find the maxes/mins? For example, once I get something like SA(x)= x^2+(8/x), what would I do from there?]

Question

I have an optimization question that was mostly answered, can someone answer the rest of it really quickly?

[When doing optimization, how would you find the maxes/mins? For example, once I get something like SA(x)= x^2+(8/x), what would I do from there?]

anonymous · Answer

Depends on what step you're at. Is this after substitution? After differentiating? Where are we at in the process?

anonymous · Answer

I've been given a problem where I have to write the surface area of a box as a function of the length of the base. I got the equation I posted above (which is correct, by the way, so no worries about that) and that's all I've done with it.

anonymous · Answer

So I assume you got that after some sort of substitution then. Given that, now you need to differentiate it and find the critical points of that derivative.

anonymous · Answer

I did that, and I got 0 and $$\sqrt{8}$$... What do I do now?

anonymous · Answer

Those are the critical numbers I got.

anonymous · Answer

So, if you differentiate what you have, you get:
$$2x - \frac{ 8 }{ x^{2} } \implies \frac{ 2x^{3}-8 }{ x^{2} }$$With that, we can get critical points from the numerator and denominator. The undefined values from the denominator just give 0, so that doesn't do us much good. The numerator however gives us:
$$2x^{3}-8 = 0 \implies x^{3} = 4 \implies x = 4^{1/3}$$. So I get that as a critical point.

anonymous · Answer

Oh, okay. What would I do after that? Would I plug it into the original equation?

anonymous · Answer

Correct. And since it is the only critical point that gives us a realistic answer, it will give the max/min.

anonymous · Answer

How do I know which one it is?

anonymous · Answer

Well, since there is only one possible answer, you can think of it as both. It's the only dimensions that make the problem work, so there is no greater answer and there is no lesser answer. It's just the answer. Up to you if you want to call it a max or a min, it can be both.

anonymous · Answer

And also, for any similar problems, assuming I get more than one critical number, the smallest number would be the min, and the larger one would be the max, correct? I wouldn't have to worry about global/local max/mins, right? (sorry about all the questions!)