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# Optimization Using the Second Derivative Test - Concept

###### 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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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

Thiswas EXCELLENT! I am a math teacher and have been looking for an easy/logical way to explain the lateral area of a cone to my students and this was incredibly helpful, thank you very much!”

I just learned more In 3 minutes of polygons here than I do in 3 weeks in my math class”

Hahaha, his examples are the same problems of my math HW!”

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