Implicit Function Homework: Finding Min/Max

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SUMMARY

The discussion focuses on determining the extrema of an implicit function and identifying whether these points are minima or maxima. The user has successfully found the first derivative and the extrema points, adhering to the condition F(a) = b. To ascertain the nature of these points, the user contemplates using the second derivative test, questioning how to derive the second derivative for an implicit function. The solution involves applying the same principles used for the first derivative to find the second derivative.

PREREQUISITES
  • Understanding of implicit differentiation
  • Knowledge of first and second derivative tests
  • Familiarity with calculus concepts related to extrema
  • Ability to interpret implicit functions
NEXT STEPS
  • Learn techniques for implicit differentiation of higher-order derivatives
  • Study the application of the second derivative test in detail
  • Explore examples of finding extrema for implicit functions
  • Review calculus textbooks or resources focusing on implicit functions and their properties
USEFUL FOR

Students studying calculus, particularly those tackling implicit functions and extrema analysis, as well as educators seeking to clarify these concepts in a classroom setting.

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Homework Statement



Hey.
I have this problem with the second part of the question.
In the first part, I needed to find all the extrema points of this implicit function and the second one is to find if it's minimum or maximum.
I posted only the derivative of the implicit function and the points I found.
BTW there was another condition to find the point, they said F(a) = b, so together with the derivative, I found the points.
Back to the question, as I was saying, the second part is to find if it's minimum or maximum, how do I do that?
I thought about finding the second derivative and plugging the points into it, but how do you find the second derivative of an implicit function, same as you find the first one?

I didn't posted the question, cause it's kind hard to translate, so I hope I gave you all the info you need.


Homework Equations





The Attempt at a Solution

 

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Use the second derivative test: If y"(x)>0 x is a minimum, if y"(x)< 0 x is a maximum.
 

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