If f(x)=x3x2−25, what are the points of inflection, concavity and critical points?

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Answer: Critical numbers: 0 and ±75Step 1Concave down on (−∞,−5) and on (0, 5)Concave up on (-5, 0) and on (5,∞)Inflection point (0, 0)Step 2f′(x)=x2(x2−75)(x2−25)2f&#039;(x) DNE at x=±5 but those are not in the domain, so they are not critical.f′(x)=0 at x=0 and at x±75 which are in the domain, so they are all critical numbers.fx)=50x(x2+75)(x2−25)3 could change sign at 0 and at ±5.On (−∞,−5),fx)&amp;lt;0, so f is concave down.On (−5,0),fx)&amp;gt;0, so f is concave up.On (0,5),fx)&amp;lt;0, so f is concave up.On (5,∞),fx)&amp;lt;0, so f is concave up.The concavity changes at x=−5,0 and 5. An inflection point is a point of the graph where concavity changes. Since -5 and 5 are not in the domain of f, there are no IPs there, but f(0)=0, so (0, 0) is an Infle pt.

If f(x)=\frac{x^{3}}{x^{2}-25}, what are the points of inflection, concavity and

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