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James Stewart

8 Edition

Chapter 4, Problem 79

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Prove Fermat’s Theorem for the case in which $f$ has a local minimum at $c$.

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06:13

02:37

Problem 3 (4 points): Prove the following Fermat’s Theorem: If f has a local minimum at C, and if f'(c) exists, then f'(c) = 0. Hint: We proved the case when f has a local maximum at c in Lec7.4. The proof for the case when f has a local minimum is very similar.

01:08

Prove the part of Theorem 3.3 that was not proved in the reading: If a function $f$ has a local minimum at $x=c$, then either $f^{\prime}(c)$ does not exist or $f^{\prime}(c)=0$.

04:52

Prove the part of Theorem $3.3$ that was not proved in the reading: If a function $f$ has a local minimum at $x=c$, then either $f^{\prime}(c)$ does not exist or $f^{\prime}(c)=0$.

01:14

Show that if $f$ has a local minimum at $c,$ then either $f^{\prime}(c)=0$ or $f^{\prime}(c)$ does not exist.

Transcript

we are going to prove Fermat’s theory for the case in which the function F has a local minimum at sea. So we know that in the text book, it was proved the theorem for the case in which the functions has local maximum. And we got to remember that the theorem says that if the function has a derivative at the point Where it has local maximum and the serial zero. So um we want to prove that the S. F. Has local minimum. And see then conservative At that point if it exists is zero. But for these we’re gonna use exercise 78. Which says that if a function F has a local minimum at the point C then G F X defined as negative Fx has local maximum value at team at the same point. So here we’re in that case because we are supposing that F has a local minimum at sea. That’s the hypothesis for the exercise 78. So, directly from exercise 78 we know that the function G. Fx define its negative F. Of X has local maximum at sea. Mhm. Okay. So now because she dysfunction has a local maximum at sea, we can apply the part of the firm and theory. That was proved in the textbook. That is the case of the local maximum at a point. And we know the function G has local maximum at a point C. So you send him and part of the theory. Okay, proof into text. Mhm. We know that G. Tell you that in that C. Is zero if you exist. So if the function…