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# What are the points of inflection for f'(x)=(3x(x^2+27))/(x^2-9)^3

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What are the points of inflection for f"(x)=(3x(x^2+27))/(x^2-9)^3

Nov 28, 2018

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$$f''(x) = \dfrac{3x(x^2+27)}{(x^2-9)^3}$$

$$\text{a point }x \text{ is an inflection point of }f(x) \text{ if both of the following hold}\\ \text{1) }f''(x)=0 \\ \text{2) }sgn(f''(x-\epsilon)) \neq sgn(f''(x+\epsilon)),~\epsilon > 0 \text{ but small}$$

$$\text{It's pretty clear that the only real value for which }f''(x)=0 \text{ is }x=0\\ sgn(f''(-\epsilon)) = 1\\ sgn(f''(\epsilon))=-1\\ \text{So }x=0 \text{ is indeed an inflection point and the only one.}$$

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Nov 28, 2018