# Blog

## How to Solve Optimization Problems in Calculus

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Anonymous
7 months ago

What a phenomenal explanation, you just saved my ap calculus grade haha

Editor
7 months ago

We’re happy to have helped. Good luck with your exam! : )

Anonymous
2 years ago

Man, if only everyone was a thorough in their explanations as this one is

Editor
2 years ago

Thank you for the compliment. We are very happy to have helped! : )

Chris M
3 years 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?

Editor
3 years 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
4 years ago

what problems can help to solve optimization

Editor
4 years ago

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

We hope that helps!

Anonymous
4 years 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

Editor
4 years 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!

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