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### Course: AP®︎/College Calculus AB > Unit 5

Lesson 6: Determining concavity of intervals and finding points of inflection: graphical# Inflection points (graphical)

Sal analyzes the graph of a function g to find all the inflection points of g.

## Want to join the conversation?

- I'm confused about the slope increasing and decreasing. Graphically, it looks like the slope changes signs around -3,0, and 3. Instead, the video showed the points of inflection in the middle of the slopes. Could you explain why that is in more depth?(35 votes)
- We are now caring about the slope of derivative. Or in other words, how the derivative of the derivative behaves.

Try changing "t" variable at the left here https://www.desmos.com/calculator/anqtkx2xtg

The red graph is your function (you can play with it)

The blue graph is your derivative

The green line is the slope of your derivative(7 votes)

- how does it become apparent that a slope is increasing or decreasing?(15 votes)
- An interesting trick that one can use for this is to draw the graph of the first derivative. Then identify all of the points in say f'(x) where the slope becomes zero. These points, where slope is zero are the inflection points. Instead of microanalyzing the graph for increasing or decreasing this is much more accurate and rigorous.(17 votes)

- i noticed something, as the slope increases the angle of the slope goes anti-clockwise and as the slope decreases angle of the slope goes clock-wise. And the point at which the 'clock' starts moving from anti-clockwise to clock-wise or vice versa, it is called point of inflection. Has anybody else noticed this?(11 votes)
- That's a really neat observation. it also links up with what positive and negative angles are, in relation to positive and negative slope.(3 votes)

- At3:15, Sal states and writes there are 3 inflection points. I see 4 inflections points. Didn't he miss an inflection point at x = - 3.5? Over the interval (-4, -3.5) I estimated a slope of at least 18 and a slope of 10 over (-3.5, -3) ===> the slope is decreasing as the function enters a concave downward. Can anyone provide a counter argument?(8 votes)
- I don't know how to find the exact point.1:47even sal says "around here"(4 votes)
- This is just finding inflection points graphically. To find the exact point you need the equation of the function and find the 2nd derivative. That will be explained in a later video. This seems to be just to get you comfortable with the concept of an inflection point.(5 votes)

- So between any 2 critical points there is always an inflection point?(4 votes)
- How exactly do you identify whether or not the slope is increasing or decreasing?The inflection point around the x value of -2 doesn't seem to have a change in slope from decreasing to increasing.(3 votes)
- The first derivative is the slope. when the derivative is 0 (a crititical point) the slope is 0.

The second derivative is the slope of the slope, or in other words if the slope is increasing or decreasing. when the second derivative is 0 (a critical point) the slope of the slope is 0, or in other words the slope goes from increasing to decreasing or the other way around. this does not mean the slope goes from positive to negative though (or the other way around.

You can have an increasing slope that is still negative, if the slope goes from -4 ot -3 it is still negative but increasing.

Let me know if that doesn't make sense.(2 votes)

- What if f'' changes sign at a point but the double derivative doesn't exist at that point, like a cusp in f'? The concavity changes at the point so would that still be considered a point of inflection?(3 votes)
- Yes it would, assuming that the function is defined at the point.

An inflection point only requires:

1) that the concavity changes and

2) that the function is defined at the point.

You can think of potential inflection points as critical points for the first derivative — i.e. they may occur if`f"(x) = 0`

**OR**if`f"(x)`

is undefined. An example of the latter situation is`f(x) = x^(1/3)`

at`x=0`

.

Relevant links:

http://depts.gpc.edu/~mcse/CourseDocs/calculus/concavity-inflectionOct12.pdf

https://math.stackexchange.com/questions/402459/an-inflection-point-where-the-second-derivative-doesnt-exist(2 votes)

- An interesting trick that one can use for this is to draw the graph of the first derivative. Then identify all of the points in say f'(x) where the slope becomes zero. These points, where slope is zero are the inflection points. Instead of microanalyzing the graph for increasing or decreasing this is much more accurate and rigorous.(3 votes)
- what does inflection mean?(2 votes)
- An inflection point is wherever a function changes direction.(2 votes)

## Video transcript

- [Voiceover] We're told let g be a differentiable function defined over the closed interval
from negative four to four. The graph of g is given
right over here, given below. How many inflection points
does the graph of g have? So let's just remind ourselves,
what are inflection points? So inflection points are
where we change concavity. So we go from concave, concave upwards, upwards,
actually let me just draw it graphically. We're going from concave
upwards to concave downwards, or concave downwards to concave upwards. Or another way you could think about it, you could say we're going
from our slope increasing, increasing, increasing, to our slope decreasing. To our slope decreasing, or the other way around. Any points where your
slope goes from decreasing, our slope goes from
decreasing, to increasing. To increasing. So let's think about that. So as we start off right over here, at the extreme left it's seems like we have a very high slope. It's a very steep curve, and then it stays increasing
but it's getting less positive. So it's getting a little bit, it's getting a little bit flatter, so our slope is at a very high
level but it's decreasing. It's decreasing, decreasing, decreasing, slope is decreasing, decreasing even more. It's even more and then
it's actually going to zero, our slope is zero and
then it becomes negative. So our slope is still decreasing. Then it's becoming more
and more and more negative, and then right around, and
then right around here, it looks like it starts
becoming less negative, or starts increasing. So our slope is increasing increasing, it's really just becoming
less and less negative, and then it's going close to zero, approaching zero, it looks like our slope is zero right over here, but then it looks like right over there our slope begins decreasing again. So it looks like our
slope is decreasing again. So it looks like our slope is decreasing. It's becoming more and more
and more and more negative, and so it looks like
something interesting happened right over there, we
got a transition point, and then right around here,
it looks like it starts, the slope starts increasing again. So it looks like the
slope starts increasing. It's negative but it's becoming less and less and less negative
and then it becomes zero and then it becomes positive and then more and more and more and more positive. So inflection points are where we go from slope increasing to slope decreasing. So concave upwards to concave downwards, and so slope increasing was
here to slope decreasing, so this was an inflection point, and also from slope decreasing
to slope increasing. So that's slope decreasing
to slope increasing, and this is also slope
decreasing to slope increasing. So how many inflection points
does the graph of g have that we can see on this graph? Well it has three over the interval that at least we can see.