00:01
In this problem, we want to draw a graph with the given conditions.
00:03
So here we have first that f prime of minus 2 is equal to 0, and f prime of 3 is equal to 0, which means that we have two horizontal tangents for f at this point.
00:20
So let's label on our graph for f of x the point minus 2 and the point 3.
00:37
Next, we have that our function f is increasing between minus 2 and 3.
00:44
Yet decreasing on minus infinity and minus 2.
00:49
So here our curve within this interval is increasing, and in this region too, but decreasing in this region.
01:14
We have that f prime prime of minus 1 over 3 is equal to f prime prime of 3 is equal to 0.
01:20
So these correspond to inflection points to a change on cavity, as opposed to before we had critical points.
01:42
Next we have our curve is concave up between minus infinity and minus 1 over 3, and 3 to infinity.
01:52
3 to infinity, our curve is concave up.
01:55
This is what concavity looks like.
02:02
And between minus infinity and minus 1 over 3.
02:06
So this here is roughly minus 1 over 3.
02:15
And so within this region, within this region, our curve is concave up, as well as this region.
02:40
And concave down between minus 1 over 3 and 3.
02:47
Concave down is the frown, and concave up is this interface.
02:54
So then let's draw a graph that satisfies all of these properties.
03:00
So for a function to be decreasing and concave up, we need to be at this left portion of the concavity.
03:08
So between minus infinity and minus 2, our curve looks like this...