00:01
In this problem, we are given the graph of the derivative of a function f, which is defined for x ranging between 0 to infinity.
00:11
We want to use this graph to answer the following questions.
00:14
Question a, we want to find the intervals for which our function f is increasing or decreasing.
00:22
To answer this question, we need to understand that when a function has a positive slope, it means that the function is increasing.
00:29
When a function has a negative slope, it means that the function is decreasing.
00:35
The slope of a function is given by its first derivative, which means that in our graph, regions for which f prime of x is positive corresponds to an increasing function, so increasing f.
00:58
If f prime of x is negative, then this corresponds to our function f that is decreasing.
01:12
Let's look at our graph here and look for regions for which f prime is positive or negative.
01:21
We see here that f prime of x is positive for x ranging between 0 to 2, for x ranging between 4 to 6, and for x ranging between 8 to infinity, which means that we're going to have three intervals for which f is increasing, which again correspond to x ranging between 0 to 2, union, x ranging between 4 to 6, union, x ranging between 8 to infinity.
02:02
Now let's find the regions for which f is decreasing, which correspond to regions for which f prime of x is negative.
02:12
We have two regions where f prime of x is negative.
02:15
We have x ranging between 2 to 4, and x ranging between 6 to 8.
02:21
So we have that f is decreasing for x ranging between 2 to 4, union, x ranging between 6 to 8.
02:35
In our next question, we want to find the values of x for which we have a local maximum or minimum in f, but we are given the graph of f prime and not f.
02:48
So again, to answer this question, we need to recall that regions for which f prime of x is equal to 0 correspond to critical points in our function f.
03:05
And critical points generally indicate a local and extreme point in our function f.
03:13
And we can differentiate between local maximum or minimum by looking at the second derivative.
03:19
So if f prime prime of x, where x is a critical point, is negative, then we have a local max.
03:34
If f prime prime of x, second derivative, of f evaluated at a critical point is positive, then we have a local minimum.
03:49
So let's first start by determining our critical points, and then we will conduct our second derivative test.
03:58
Critical points will be labeled in black, and we see here that we have three locations for which f prime of x is equal to 0.
04:07
We have x equal to 2, x equal to 4, and x equal to 8.
04:16
So we have three critical points.
04:18
Now i want to determine whether these critical points are maximum or minimum.
04:22
And to do so, we need to look at the sign of the second derivative.
04:27
Or we can look at the slope of our first derivative, because the slope of our first derivative corresponds to our second derivative.
04:37
So looking at our first critical point here, we have a negative slope.
04:43
Then we have a positive slope.
04:47
And then we have again a positive slope.
04:50
So if we have a negative slope here, it means that f prime prime of x is negative.
05:01
And if we have a positive slope here, it means that f prime prime is positive...