00:01
Hey there, welcome to numerate.
00:02
We are asked here to sketch a graph for our following function in which we would have to find our critical points first.
00:12
So let's go ahead and find the derivative of the function.
00:15
So this is f prime of x which is equivalent to 3x squared plus 6x plus plus 3.
00:27
So hence what we have here when we set it equal to 0 we have 3 times x squared plus 2x plus 1 which is equivalent to 0.
00:41
So with this we will get an x here that is equivalent to negative 1 which is treated as a repeated root.
00:51
So we have x equals negative 1.
00:55
Now to find the concavity, this will be representing the critical point.
01:02
So find critical points.
01:12
And now we're going to find the concavity.
01:19
So to find the concavity, we are going to be taking the derivative again.
01:24
So we're going to differentiate again.
01:26
Again.
01:26
So hence, f double prime of x is equivalent to 6x plus 6, in which we have this being x equals negative 1.
01:38
So this signifies that since we have negative 1, we can see that it's going to be concave up to the right and concave down to the left, so which passes through negative 3 on the y -axis.
01:55
So hence, these are the following things we will observe, that we will have a critical point.
02:05
So we have a critical point at x equals negative 1.
02:16
We will also observe here that it will be concave up for when we have x being greater than negative 1, and concave down for 4 when x is, so when 4x is less than negative 1.
02:46
And finally, we will have the point, or it will pass through, so this function will pass through negative 3 on the y -axis.
03:00
So basically, y equals negative 3.
03:06
So hence, let's go ahead and sketch our graph here.
03:10
So we will have our x and y -axis is here.
03:20
Alright, let's go ahead and graph our points...