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

Start by taking the second derivative..

...which is simpler to multiply by:

...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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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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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.

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.

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.

- How do you find all points of inflection of the function #f(x)=x^3-3x^2-x+7#?
- How do you find points of inflection of #(1-x^2)/x^3#?
- For what values of x is #f(x)= (5x-x^3)/(2-x)# concave or convex?
- What are the points of inflection of #f(x)=x+sinx # on the interval #x in [0,2pi]#?
- Is #f(x)=(x-2)^3-x^4+x# concave or convex at #x=0#?

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