The only point at which f″(x)=0 or is undefined (f′ is not differentiable) is at x=0. If x<0, then f″(x)<0 so f is concave down. If x>0, then f″(x)>0 so f is concave up. At x=0 the concavity changes so the point (0,f(0))=(0,0) is an inflection point of f(x)=x3.

How do you know when concavity changes?

To find when a function is concave, you must first take the 2nd derivative, then set it equal to 0, and then find between which zero values the function is negative. Now test values on all sides of these to find when the function is negative, and therefore decreasing.

At what point does concavity change?

Inflection points are where the function changes concavity. Since concave up corresponds to a positive second derivative and concave down corresponds to a negative second derivative, then when the function changes from concave up to concave down (or vise versa) the second derivative must equal zero at that point.