# Blog

## How to Solve Optimization Problems in Calculus

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Chris M
11 months ago

Nice explanation and methodical approach. Setting up the problem is 99% of the problem. I’m still trying to figure out on other optimization solutions what yo do if the 2nd derivative is simply a constant. If f ‘(x) has a desired min or max, and f ‘’(x) differentiates to a constant, what does this mean to the max or min in the first derivative test? Does the sign of the constant alone in f ‘’(x) then determine concavity since there’s no potential inflection point? If that constant’s sign is negative (concave down) and the solution requires an optimization max, does that satisfy a proof? And vice versa if concavity is positive (concave up) confirm a minimum in the first derivative test?

Matheno
Editor
11 months ago

Thanks, Chris. We’re glad to know you liked our explanation and approach. And agreed about getting the problem set-up right as the vast majority of the work here.

The answer to all of your questions is: yes! If the second derivative is a negative constant, then the function is concave down everywhere, and so you’re guaranteed that the point x=c you found where f'(c) = 0 is a maximum. (See the figure below.) Similarly, if the second derivative is a positive constant, then the function is concave up everywhere, and so the point x=c where f'(c) = 0 is guaranteed to be a minimum. And the fact that there’s no point of inflection anywhere doesn’t affect those conclusions.

The only thing that you wrote that isn’t quite right are the very last words, “in the first derivative test”; instead, you’re using the Second Derivative Test. That test is just as conclusive as the First Derivative Test, and is often easier to use. The one exception is if the second derivative is zero at the point of interest (f”(c)=0), in which case the Second Derivative Test is inconclusive and you have to revert to the First Derivative Test. But otherwise, the conclusion you reach with the Second Derivative test is indeed conclusive.

Hope that helps, and thanks for asking!

Anonymous
1 year ago

what problems can help to solve optimization

Matheno
Editor
1 year ago

Thanks for asking! We have more completely solved optimization problems on this page: Optimization: Problems and Solutions.

We hope that helps!

Anonymous
1 year ago

very nicely organized! however i think it would have been more effective with some numbers, instead of variables. it can get hard to follow, especially when there’s multiple(in this case). but it was still lovely and easy to follow

Matheno
Editor
1 year ago

Thank you for your nice comment, and for your suggestion. We’ll keep it in mind for future posts. For now: thanks very much!

Jon
1 year ago

I am having a tough time differentiating what it means when they say minimize(or maximize) a situation, as in how do the answers vary one from the other? I understand all the steps. I just want to be sure I pick the right “direction” when presented with an optimization problem….

Matheno
Editor
1 year ago

We’re happy to try to answer your question. First, we’d like to ask for a bit of clarification so we can be sure we address the question you actually have. Can you say a bit more about “how do the answers vary one from the other” ? Even better, could you give us an example, or two, of a problem where you don’t know what direction to pick initially? Is it that you’re not sure whether you need to maximize or minimize a particular quantity, or that when you’re presented with a situation you don’t know where to begin at all, or ??

And we know it’s tough to ask a clear question when you’re learning a new topic, so please just add whatever you can.

Thanks very much, and thanks again for posting your question, which I’m sure other students have as well!

Anonymous
2 years ago