MHB Finding Max & Min of Optimization Problem: How To Distinguish

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Finding the maximum and minimum of an optimization problem involves identifying critical values where the first derivative equals zero. The first derivative test helps determine whether these critical points are relative maxima or minima by analyzing the sign changes of the derivative. The second derivative test further confirms the nature of these points; if the second derivative is positive at a critical point, it indicates a local minimum, while a negative value indicates a local maximum. Both tests are essential for distinguishing between different types of extrema in optimization problems. Understanding these tests is crucial for effectively solving and analyzing optimization equations.
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How are Finding the max and min of a optimization problem different. and how do you distinguish them in an equation?
 
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Finding the critical values is the same, and then we can use either the first or second derivative tests to determine the nature of the extremum, or whether it is actually an extremum or not.

Are you familiar with these two tests?
 
Yes I am familiar
 
thinkbot said:
Yes I am familiar

Can you briefly explain how they work? :D
 
find f^1(x) = 0 x('s)= critical numbers plus some points (a,b) or [a,b]
f(x,a,b) = extrema Larget # max smallest # min
Using f^2(x) for the critical #'s f2(x) > 0 then min and ect.
 
thinkbot said:
find f^1(x) = 0 x('s)= critical numbers plus some points (a,b) or [a,b]
f(x,a,b) = extrema Larget # max smallest # min
Using f^2(x) for the critical #'s f2(x) > 0 then min and ect.

I meant can you explain how and why the first and second derivative tests for relative extrema work, i.e., the rationale behind them.
 
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