00:01
So this question is giving me the function f of x equals negative 4 over 1 plus 2x.
00:11
And i want to find the intervals where i'm concave up.
00:15
So i need to take the second derivative.
00:17
So let's take the first derivative first.
00:19
Use the quotient rule.
00:19
So it says keep the bottom, take the derivative of the top, which is zero, minus, keep the top, take the derivative to the bottom, which is just two.
00:30
All over the bottom squared.
00:35
I'm going to simplify that.
00:36
I'm going to get 8 over 1 plus 2x squared.
00:43
And now i'm going to take the derivative again.
00:46
So second derivative would be keep the bottom, take the derivative of the top.
00:54
Again, zero minus.
00:56
Keep the top.
00:59
Take the derivative of the bottom.
01:00
Using the chain rule.
01:02
So the two comes down in front.
01:03
I leave the inside alone.
01:05
Drop it down in degrees, so it's 1 times the derivative of the inside, which is 2, all over the bottom, when i square something that's raised to the second power already, it becomes the fourth power.
01:20
This cancels out over here, times 0...