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Optimization Using the Second Derivative Test - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Some optimization problems can be solved by use of the second derivative test. If the second derivative is always positive, the function will have a relative minimum somewhere. If it is always negative, the function will have a relative maximum somewhere. Other ways of solving optimization problems include using the closed interval method or the first derivative test.

We have one more optimization method to talk about and it's the second derivative test for absolute max and min. Let me show you how it works, suppose you have a function of y equals f of x and that function you know has a second derivative which is always positive everywhere on it's domain. Well if it has a critical point and that critical point will be an absolute minimum it's pretty much guaranteed. So that tells us how second derivative test basically works, you start with the continuous function on some interval i that's its domain and suppose f prime of c is 0 that is one critical point.
If the second derivative is always positive on domain then f will have an absolute minimum so think second derivative is positive it'll be shaped like this, there will be a minimum at x=c and if the second derivative is always negative on the interval it'll have an absolute maximum at x=c that's the second derivative test. Remember the second derivative has to be always positive or always negative.

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