00:01
Okay, so we need to do the second derivative test for this x e to the negative x squared.
00:08
And what we need to do is first of all find the derivative, which is a product rule.
00:13
There's a product right here.
00:14
So i like to do the derivative of x, leave e to the negative x squared alone.
00:18
And now you leave the x alone, you do the derivative of e to the negative x, which is itself.
00:24
But then you have to multiply by the derivative of negative x squared, which would be negative 2x.
00:29
So what i would do is factor out this e to the negative x squared.
00:34
You're left with 1 minus 2x squared in there.
00:39
Now this little bit will always be positive, so we don't have to worry about that.
00:44
But 1 minus 2x squared could equal zero.
00:48
So i can add that over, divide, and then square root.
00:53
Now i'll just leave it as positive or negative the square root of one half or 1 over root 2.
01:00
And then we need the second derivative.
01:04
So i'm going to look at my blue equation and do the second derivative, which is another product rule where i would leave the e to the negative x squared or take the derivative of that, negative 2x.
01:18
Chain rule again, leaving the right side alone.
01:23
And now leave e to the negative x squared alone, taking the derivative of the right side, which would be negative 4x.
01:29
And again, i can factor, but at this point i probably need to distribute.
01:36
Like negative 2x, if i add this negative 4x to it, i have negative 6x.
01:40
And then negative 2x times negative will be a positive 4x cubed.
01:47
And now what you have to do for the second derivative test is see what happens if you plug in the 1 over root 2.
01:55
Now this piece will be positive still.
01:59
And let's just go to a calculator.
02:01
Although, yeah, negative 6 times 1 over root 2 plus 4 times 1 over root 2 cubed gave me an answer of negative 2 .828...