Extrema in Several Variables: A different way?

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SUMMARY

The discussion centers on finding extrema in several variables, comparing the first derivative test with the second derivative test. The first derivative test is highlighted as a simpler method that involves determining the sign of the first derivative across intervals, while the second derivative test requires calculating second derivatives. The user inquires about alternative methods for identifying extrema in multivariable calculus, expressing a desire for a quicker approach similar to the first derivative test used in single-variable calculus.

PREREQUISITES
  • Understanding of single-variable calculus, specifically the first and second derivative tests.
  • Familiarity with multivariable calculus concepts, particularly partial derivatives.
  • Knowledge of the determinant test for classifying critical points in several variables.
  • Ability to interpret sign changes on number lines for determining extrema.
NEXT STEPS
  • Research the application of the first derivative test in multivariable functions.
  • Explore alternative methods for finding extrema in several variables, such as the Hessian matrix.
  • Study the geometric implications of partial derivatives in optimization problems.
  • Learn about critical points and their classification using the second derivative test in multiple dimensions.
USEFUL FOR

Students and educators in calculus, particularly those focusing on optimization in multivariable functions, as well as mathematicians seeking efficient methods for finding extrema in several variables.

nth.gol
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Hi,

In high school, I was shown an unconventional but quicker way to find max/mins. I'm not sure how common it is but we did it because we learned to curve sketch without calculus first.

Take f'(x) =0, and solve for the roots. Construct a number line and place all roots on the number line. Alternate + - from the right, unless there is a negative out front, and don't change signs around squared roots.

From here you extrapolate max mins based on sign. This is all nice and dandy compared to the first derivative test.

In several variables, however, I am currently being taught to use the partial second derivative or Determinant test.
Is there a better way? A quicker one like this? I understand geometrically the implications of the second partial test, and the cases where the pure and mixed partials affect the type of extrema. But is there a similar test to that in single variable?

Thank you
 
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That is, in fact, what is known as the "first derivative test". It is typically given in most Calculus texts just before the "second derivative test". It is simpler than the second derivative test in that you do not need to find the second derivative. However, it also requires that you be able to find the sign of that first derivative on intervals rather than at individual points as with the second derivative test. Which is easier really depends upon the function.
 
HallsofIvy said:
That is, in fact, what is known as the "first derivative test". It is typically given in most Calculus texts just before the "second derivative test". It is simpler than the second derivative test in that you do not need to find the second derivative. However, it also requires that you be able to find the sign of that first derivative on intervals rather than at individual points as with the second derivative test. Which is easier really depends upon the function.

Im not sure you really answered my question, but thanks for the input.
Is there another test in several variables?
 

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