Finding inflection points: 2015 AP Calculus AB 5c
At x = -1 the narrator says the point of inflection is 'when f prime of x is zero'; he means f'' is zero (horizontal tangent on f'). A correcting note was added.
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The video finds the points of inflection of f from the graph of f'. Inflection points are where concavity changes, which means f'' changes sign, which means f' switches between increasing and decreasing. On this graph that happens at x = -1, x = 1 and x = 3.
You'll be able to
- define a point of inflection as where concavity changes.
- explain that f has an inflection point where f' changes from increasing to decreasing or vice versa.
- find inflection points of f from a graph of f'.
Key words
- point of inflection
- A point where a graph changes from concave up to concave down, or the other way round.
- concave up
- Curving like a bowl that holds water; happens when f'' > 0.
- f prime prime (f'')
- The second derivative; its sign tells you the concavity of f.
Transcript
- [Voiceover] So part c. Find the x-coordinates of all points of inflection for the graph of f. Give a reason for your answer. So points of inflection, points of inflection happen when we go from concave upwards, concave upwards to downwards or vice versa. So this is true if and only if f prime prime of x goes from positive to negative or vice versa.
So where do we see f prime 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. F prime goes from increasing to decreasing or vice versa. We're seeing a lot of vice versa here.
So now let's, and I want you to think of it in terms of f prime because we have the graph of f prime. So 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 zero. That's because a prime is differentiable so the derivative is definitely, the derivative is zero right at that point of inflection right over here. So with that happens at x equals negative one.
And over here, then f prime starts increasing but then right at x equals one then it starts decreasing. So at x equals one, we have another point of inflection. That's where we have that zero, that zero of tangent line with slope zero. And then we're decreasing, decreasing, decreasing, decreasing, decreasing, increasing.
All right, so this is going to be another point of inflection, x equals three. So these are at three points of inflection. So this happens, happens at x equals negative one, x equals one and x equals three. So these three points on our graph of f prime where we see f prime goes from decreasing to increasing or increasing to decreasing or decreasing to increasing.
All right, all right. Now, well, I'll do the last part on the next video.
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