00:02
Okay, so we have a function, that's what that is, is equal to e to the 2x plus e to the negative x.
00:10
And we need to find the connection points, concavity, increasing and decreasing critical points and inflection points.
00:17
So that means we need to first take a derivative.
00:20
The first derivative is going to give us 2e to the 2x minus e to the x, excuse me, e to the negative x.
00:31
We can rewrite this as 2e to the 3x minus 1 over e to the x.
00:41
And we need this to be equal to 0 because we're trying to find critical points.
00:45
And so that brings down to 2e to the 3x minus 1 equaling 0.
00:51
That gives us e to the 3x is equal to positive 1 half, which is good, e is only greater than 0.
01:01
And so when we take the natural logarithm of both sides, we'll get 3x is equal to natural logarithm of 1 half.
01:09
And so that means this is going to be equal to a third of the natural logarithm of 1 half.
01:15
That's only one critical point.
01:17
But this has a critical point, so this would be.
01:21
So we'll test some points.
01:27
We'll see, for example, a little bit less than this f trim of negative 1.
01:33
From a negative 1 is going to be equal to 2 divided by e squared minus e.
01:42
That'll be less than 0.
01:45
So we'll be decreasing, decreasing on negative infinity to the 1 third natural logarithm of 1 half.
01:59
And so let's test a point larger than the critical point.
02:03
So if i put in 1, i'm going to get 2e squared minus 1 over e squared is greater than 0.
02:15
And so it's increasing on the interval.
02:23
Increasing on the interval, critical point, 1 thing like that, so logarithm of 1 half to infinity.
02:34
So that is increasing and decreasing.
02:39
Now, the second derivative is going to be equal to 4e to the 2x plus e to the negative x, which will be equal to 4e to the 3x plus 1 over e to the x...