Good dayCan anyone pls teach me how to find the extrema of this

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    Extrema
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SUMMARY

The discussion focuses on finding the extrema of a general second-degree polynomial function, specifically f(x,y) = x² + 4xy + 4y² - 2x + 3y + 6. To determine the extrema, one must calculate the first partial derivatives, set them to zero to find critical points, and then use the second partial derivatives to classify these points. The process is not covered in the participant's syllabus, indicating a gap in their current mathematical education.

PREREQUISITES
  • Understanding of partial derivatives
  • Familiarity with second-degree polynomial functions
  • Knowledge of critical points in multivariable calculus
  • Ability to apply the second derivative test for classification
NEXT STEPS
  • Study the method for calculating first partial derivatives of multivariable functions
  • Learn how to find critical points for functions of two variables
  • Research the second derivative test for classifying extrema
  • Review examples of extrema in textbooks or online resources
USEFUL FOR

Students studying calculus, particularly those tackling multivariable functions, as well as educators seeking to enhance their teaching materials on optimization techniques.

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Good day!

Can anyone pls teach me how to find the extrema of this function (general second degree polynomial function): f(x,y)=x^2+4xy+4y^2-2x+3y+6. actually, my problem is i don't know how to find the extrema of functions of this form f(x,y)=ax^2+bxy+cy^2+dx+ey+f.

Your help will be highly appreciated.

Thanks!
 
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Find the two first partials and set them to zero. The critical points (x, y) are those for which both first partials are zero. There's a test for categorizing the critical points that involves the 2nd partials.

Your textbook should have examples showing how this is done.
 


Thank you for the quick reply sir. I haven't read this topic from a book before. It is not covered in our syllabus. This is an open problem given to me by my professor.

I'll read more and will post again if i can't figure out the mess.
 

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