Second Derivative Test

# Second Derivative Test - Second Derivative Test Suppose...

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Second Derivative Test Suppose that is a critical point of such that and that is continuous in a region around . Then, 1. If then is a relative maximum. 2. If then is a relative minimum. 3. If then can be a relative maximum, relative minimum or neither. The proof of this fact uses the Mean Value Theorem which, if you’re following along in my notes has actually not been covered yet. The Mean Value Theorem can be covered at any time and for whatever the reason I decided to put it after the section this fact is in. Before reading through the proof of this fact you should take a quick look at the Mean Value Theorem section. You really just need the conclusion of the Mean Value Theorem for this proof however. Proof of 1 First since we are assuming that is continuous in a region around then we can assume that in fact is also true in some open region, say around , i.e. . Now let x be any number such that , we’re going to use the Mean Value Theorem on . However, instead of using it on the function itself we’re going to use it on the first derivative. So, the Mean Value Theorem tells us that there is a number such that, Now, because we know that and we also know that so we then get that, However, we also assumed that and so we have,

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Or, in other words to the left of the function is increasing. Let’s now turn things around and let
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## This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.

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Second Derivative Test - Second Derivative Test Suppose...

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