Question

The figure below shows the graph of f ', the derivative of the function f, on the closed interval from x = -2 to x = 6. The graph of the derivative has horizontal tangent lines at x = 2 and x = 4.
Find the x-coordinate of each of the points of inflection of the graph of f. Justify your answer.

Answer 1

The inflection points are when acceleration is equal to zero.  If the above graph is dy/dx, acceleration will be equal to zero when the slope of it is zero because that will mean velocity is constant at that point.

So if the graph of dy/dx has horizontal slopes at x=2 and 4, they are the x-coordinates of the inflection points of f(x).

Answer 2

The points of inflection are the points where the slope of f'(x) is zero

basically where the horizontal tangents are
at x=2 and x=4