Proofs of Derivative Applications Facts

Proofs of Derivative Applications Facts - Proofs of...

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Proofs of Derivative Applications Facts/Formulas In this section we’ll be proving some of the facts and/or theorems from the Applications of Derivatives chapter. Not all of the facts and/or theorems will be proved here. Fermat’s Theorem If has a relative extrema at and exists then is a critical point of . In fact, it will be a critical point such that . Proof This is a fairly simple proof. We’ll assume that has a relative maximum to do the proof. The proof for a relative minimum is nearly identical. So, if we assume that we have a relative maximum at then we know that for all x that are sufficiently close to . In particular for all h that are sufficiently close to zero (positive or negative) we must have, or, with a little rewrite we must have, (1) Now, at this point assume that and divide both sides of (1) by h . This gives, Because we’re assuming that we can now take the right-hand limit of both sides of this.
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We are also assuming that exists and recall that if a normal limit exists then it must be equal to both one-sided limits. We can then say that, If we put this together we have now shown that . Okay, now let’s turn things around and assume that and divide both sides of (1) by h . This gives, Remember that because we’re assuming we’ll need to switch the inequality when we divide by a negative number. We can now do a similar argument as above to get that, The difference here is that this time we’re going to be looking at the left-hand limit since we’re assuming that . This argument shows that . We’ve now shown that and . Then only way both of these can be true at the same time is to have and this in turn means that must be a critical point. As noted above, if we assume that
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Proofs of Derivative Applications Facts - Proofs of...

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