What are the points of inflection of #f(x)= x(x^2+1) #?

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Answer 1 · 2017-09-16T12:21:06

Start by taking the second derivative..

...which is easier if you multiply through:

#d/dx(x^3+x) = 3x^2 + 1#

#(d^2x)/dx^2 = 6x#

...and your inflection point is where this quantity goes from positive to negative, or vice versa.

This happens at point x = 0.

On a graph, an inflection point is where the curve shifts over from a left turn to a right turn, or vice versa:

graph{x(x^2 + 1) [-10, 10, -5, 5]}