00:01
Okay, so these functions we're going to solve is about convexity of certain functions.
00:06
I'm going to write them one by one as we go.
00:10
The first one is going to be the easiest one, e to the x minus one for x in r, just one dimension.
00:18
And of course, in all of them, these are going to be differentiable functions on the domain that we are considering.
00:22
So we're always going to use a second derivative to study convexity.
00:25
So in this case, let's just see, we have f prime of x, which is the exponential, and f prime prime of x is also the exponential.
00:34
Which is always positive.
00:36
Positive second derivative implies convex function.
00:41
Okay, and the spirit will be the same when we move on to two dimensions.
00:45
Okay, so first one is convex.
00:47
Second one, so now f of x one, x two, defined as x one times x two.
00:56
And here we have to study the second derivative.
00:59
So these of course, second matrix of second partial derivatives.
01:02
So let's start, d x one f is of course, x two.
01:07
D x two f is x one.
01:10
So our hessian matrix, the matrix of second derivatives, this is defined as d two x one x one, d two x one x two, d two x one x two, okay, i forgot the f, of course.
01:27
And then d two x two, x two with f.
01:30
In this case will be, okay, so d x one, d x one is zero.
01:36
And d x two, d x two is zero.
01:39
But now d x two, d x one, d x two, d x one is one.
01:45
And d x one, d x two is also equal to one.
01:47
So we've studied the sign of this, of the hessian function.
01:52
So what we conclude from here is that, we want to know if it is positive definite or negative definite.
01:58
Let's look, the trace of the hessian is equal to zero.
02:04
And the determinant is equal to minus one, right? so the conclusion in here is that, we have two eigenvalues.
02:16
And that one of them, you remember that the derivative is positive.
02:22
The derivative is the multiplication of the eigenvalues.
02:26
Sorry, i just lost my focus for a moment.
02:28
Determinant is the product of the eigenvalues.
02:31
And if it is negative, it must mean that one is positive and the other one is negative.
02:36
And this means that the function is neither concave nor convex, neither concave nor convex, right? so we have regions where we have one concavity and regions where we will have another.
02:54
So now c is f of x one, x two, which is one over x one and x two.
03:03
And this is for positive, positive x one and x two.
03:11
So again, we just make the calculation.
03:13
So let's get on with it.
03:16
So d x one f will be, so we're going to have minus one over x two was already there.
03:23
And then we get an x one squared and d x two f is gonna be minus one over x one was already there.
03:31
We get an x two squared.
03:33
So that's when we get the hessian now, we differentiate again in x one.
03:39
So this become two over x two was already there.
03:43
And the x one becomes cubed.
03:44
Here, same thing, two over x one was there and the x two becomes cubed.
03:49
And the cross derivatives will be this one, the minus sign will disappear.
03:55
So we're going to have a one over x one, x two, x one squared, x two squared.
04:02
And same thing in here, right? x one squared, x two squared.
04:05
So let's see what we can conclude...