Why Do the Steps to Maximize and Minimize a Quantity Align?

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SUMMARY

The steps to maximize and minimize a quantity are fundamentally aligned due to the mathematical principles governing critical points. Both processes involve finding the derivative of a function, setting it to zero to identify critical points, and using the second derivative test to determine the nature of these points. Specifically, maximizing a function f(x) is equivalent to minimizing its negative, -f(x), which highlights the symmetry in optimization techniques. The critical points where the tangent is parallel to the x-axis are common to both maximization and minimization, differing only in the sign of the second derivative.

PREREQUISITES
  • Understanding of calculus, specifically derivatives and critical points.
  • Familiarity with the second derivative test for concavity.
  • Knowledge of function behavior and optimization techniques.
  • Basic algebra skills for solving equations.
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  • Study the concept of critical points in calculus.
  • Learn about the second derivative test in detail.
  • Explore optimization techniques in multivariable calculus.
  • Investigate the relationship between maximizing f(x) and minimizing -f(x).
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Students of calculus, mathematicians, and anyone interested in optimization problems in mathematics or applied fields.

m0286
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The question is,
The steps you follow to maximize a quantity are the same as the steps to minimize a quantity, explain why?
Im not sure why?
I know to maximize or minimize a quantity, you find the derivative, let it equal 0 and solve to find the value of the variable that results in a maximum or minimum, then you find the maximum or minimum value (if required) by substitution, then check to see you have the maximum or minimum by test-second derivative if >0 then min, if <0 max. Can anyone please explain to me why? THANKS SOO MUCH!
 
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Both type of points are among the critical points.The fact that the tangent is || to the Ox axis in both situations should make the algorithms similar.And the only difference is the sign of the second derivative.

Daniel.
 
Compare minimizing f(x) with maximizing -f(x).
 

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