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Prove Fermat’s Theorem: If a local maximizer exists, then f'(c) = 0 and if a local minimizer exists, then f'(c) = 0. F(x) over the domain [a,b]. Mean Value Theorem: f”(c) = (F(b) – F(a))/(b-a). Rolle’s Theorem: F(a) = F(b) where F'(c) = 0.

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06:13

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

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.

02:21

If $f$ is differentiable on the interval $[a, b]$ and $f^{\prime}(a)<0<f^{\prime}(b),$ prove that there is a $c$ with $a<c<b$ for which $f^{\prime}(c)=0 .$ (Hint: Use the Extreme Value Theorem and Fermat’s Theorem.)

05:09

Use Fermat’s Theorem to prove that the function below has no local extrema. f(x) = 1/(âˆš(x^4+x^2))

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