Prove: Suppose f:[a,b]->R, c an interior point of [a,b], and there is an n E N f

Question

Prove: Suppose f:[a,b]->R, c an interior point of [a,b], and there is an n E N for which f and all of its nth derivatives are continuous on a neighborhood of c with f'(c)=f''(c)=...=f^(n-1)(c)=0, but f^(n)(c) does not equal 0.
I. If is even and f^(n)(c)>0, then f(c) is a relative minimum.

Explanation / Answer

suppose f: [a,b] => all real numbers , c an interior point of [a,b], and there is an n E N for which f, and all of its nth derivatives are continuous on a neighborhood of c with f'(c) = f " (c) = ... f ^(n-1)of (c) = 0 but f^(n) of (c) does not equal 0.

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