Let I C R be an open interval and let ƒ : I → R be twice differentiable. We say that c € I is an inflection point of f if ƒ" changes sign at c, that is, if for some ɛ > 0 we have ƒ"(x) < 0 if x ≤ (c − ɛ, x) and ƒ"(x) > 0 if x = (x, c + ɛ), or the other way around[ƒ"(x) > 0 if x € (c − ɛ,x) and ƒ"(x) < 0 if x ≤ (x,c+ɛ)].

Use Fermat's theorem from MATH1115 (the derivative of a differentiable function equals 0 at interior maximum and minimum values) in combination with Taylor's theorem to show that a twice differentiable function cannot have a maximum or a minimum at an inflection point.

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