Home > Calculus > Determining Points of Inflection for a Function

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

Answer 1

Charles Anctil

Start by taking the second derivative..

...which is simpler to multiply by:

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

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

...and the point at which this quantity changes from positive to negative, or vice versa, is your inflection point.

This takes place at x = 0.

An inflection point on a graph is the point at which the curve changes from a left turn to a right turn, or the other way around:

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

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Answer 2

Elijah Abram

To find the points of inflection of ( f(x) = x(x^2 + 1) ), we need to find the second derivative of the function, ( f''(x) ), and then solve for the values of ( x ) where ( f''(x) = 0 ) or ( f''(x) ) does not exist.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.