2015 AP Calculus AB 5c | AP Calculus AB solved exams | AP Calculus AB | Khan Academy

So part C: Find the x-coordinates of all points of inflection for the graph of f. Give a reason for your answer.

Points of inflection happen when we go from concave upwards to downwards or vice versa. This is true if and only if f double prime of x goes from positive to negative or vice versa.

So, where do we see f double prime of x going from positive to negative? Well, that's going to be true if and only if f prime of x goes from being increasing to decreasing or vice versa.

I'm using a lot of "vice versa" here. Now, I wanted to think of it in terms of f prime because we have the graph of f prime. f prime goes from increasing to decreasing or vice versa, or we could go from decreasing to increasing.

Let's think about it. Let's see over here: f prime is decreasing, decreasing, decreasing, decreasing, and then it increases. So we have a point of inflection right over here, right when f prime of x is 0.

That's because f prime is differentiable, so the derivative is definitely zero right at that point of inflection. Right over here, this happens at x equals negative one.

Then, f prime starts increasing, but then right at x equals one, it starts decreasing. So at x equals one, we have another point of inflection, and that's where we have that zero—a tangent line with slope zero.

Then, we're decreasing, decreasing, decreasing, decreasing, decreasing, increasing. Alright, so this is going to be another point of inflection at x equals 3.

So these are our three points of inflection. This happens at x equals negative one, x equals one, and x equals three. These three points on our graph of f prime show where f prime goes from decreasing to increasing or increasing to decreasing or decreasing to increasing.

Alright, now, well, I'll do the last part of the next video.