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### Course: AP®︎/College Calculus AB > Unit 9

Lesson 2: AP Calculus AB 2015 free response- 2015 AP Calculus AB/BC 1ab
- 2015 AP Calculus AB/BC 1c
- 2015 AP Calculus AB/BC 1d
- 2015 AP Calculus AB 2a
- 2015 AP Calculus AB 2b
- 2015 AP Calculus 2c
- 2015 AP Calculus AB/BC 3a
- 2015 AP Calculus AB/BC 3b
- 2015 AP Calculus AB/BC 3cd
- 2015 AP Calculus AB/BC 4ab
- 2015 AP Calculus AB/BC 4cd
- 2015 AP Calculus AB 5a
- 2015 AP Calculus AB 5b
- 2015 AP Calculus AB 5c
- 2015 AP Calculus AB 5d
- 2015 AP Calculus AB 6a
- 2015 AP Calculus AB 6b
- 2015 AP Calculus AB 6c

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# 2015 AP Calculus AB 5c

Inflection points for a function.

## Want to join the conversation?

- If in the exam, I write simply:

When d^2y / dx ^ 2 = 0 , x= -1 , x=1, x=3

will i get full points. It seems simpler(3 votes)- I don't think so. The question says that you need to give a reason for why you got what you got. If you don't completely answer the question, I don't think you'll get full points. In general, I'd say just answer the question thoroughly with a detailed explanation (if there's time to write one; if not, then sum it up).(4 votes)

- Maybe I'm not understanding something, but it seems to me the last part of the answer, "f'(x) goes from increasing to decreasing or vice versa" is not what he demonstrated finding on the graph. It looked to me like he was finding the points where the slope went from negative to positive or vice-versa, i.e. relative minimums and maximums, not where the slope went from increasing to decreasing. Wouldn't the slope changing from increasing to decreasing on the main function, f, represent the change in concavity?(1 vote)
- If f'(x) is increasing, its slope is positive. Just to be clear, the graph shown is a graph of f'(x), not f(x). The slope of f(x) changing from increasing decreasing would be at the point where f'(x) = 0.(1 vote)

## Video 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.