00:01
Here we're given the graph of f on the closed interval negative 4 to positive 7.
00:07
So now that we're also told that there is another function, g, which is not shown here, which is the antiderivative of f.
00:15
So another way of writing that is really to say that f of x is the derivative of g, since g is the anti -derivative of f.
00:27
Okay, so this is graph of app, or we can basically say this is the graph of g prime of x.
00:36
Okay, and we're also given that g of 1 is equal to 4, and we'll be using that later on.
00:43
So now the first part of this question is to find the x coordinate of each point of inflection of graph of g.
00:53
So the first thing you need to ask yourself is what is a point of inflection? so the point of inflection would be where the second derivative, meaning g double prime of x is equal to 0, and g double prime of x has to change signs.
01:15
Okay, so change signs.
01:21
So meaning, sorry, g double prime of x equals zero or does not exist.
01:31
Okay.
01:31
So meaning if at a particular.
01:34
Point, we have an inflection point.
01:37
That means the concavity has to change before and after.
01:42
So in other words, if we're talking about g double prime of x, this is really telling us that, well, g double prime of x is just f prime of x.
01:52
So it has to be where f prime of x is equal to zero or does not exist.
01:58
And f prime of x has to change signs.
02:05
So now looking at our function f, we can easily find what f prime of x is, right? because that's just the derivative or the slope of this function here.
02:19
Okay, so already we can see that where f prime of x is equal to zero where does not exist would be here, here, here, and here.
02:30
Right? so at these three points, f prime of x does not exist.
02:35
And then at this point here, f prime of x is equal to zero.
02:39
So now the other thing we need to see is does f prime of x change signs at any of those locations? so here, the slope before this point was negative, and then after the point it becomes positive.
02:53
So yes, it does change sign here.
02:57
Okay, what about here? well, the slope before x is equal to 1, we have a positive slope because it's going up.
03:04
And then after x equals when the slope is going down.
03:08
Okay, so again, remember you're not looking at whether f is positive or negative.
03:12
You're looking at the slope of f.
03:14
Okay, so positive slope, negative slope.
03:17
So yes, f prime of x does change sign at that point.
03:22
And now what about this point? well, before this point, we have a negative slope because it's going down.
03:27
And after this point, it's positive slope.
03:29
So again, there is a change in sign.
03:32
And now the last point of interest.
03:34
Right over here.
03:36
We have positive slope before and after the point we still have positive slope.
03:41
So there's no sign change here.
03:44
So therefore for part a we found three points of inflection for g.
03:51
So the points inflection would be x is equal to zero, x is equal to one and x is equal.
04:02
So we know that this is three because we're told that this here is a semicircle, right? and it goes from 1 to 5.
04:09
So we know that the middle point must be...