00:01
All right, so to find the maximum of this equation, first we need to determine what our boundaries are, and we're given in the problem that the boundaries are the function y equals 2x, y equals 2, and x equals 0.
00:16
So let's determine graphically what this looks like.
00:18
So on the two -dimensional plane, where this is the x -axis and this is a y -axis, this is going to look something like this, where first let's say, okay, y -equals 2 -6, so that brings up up here.
00:32
This is a line y equals two, and this is a line x equals zero.
00:36
And so we are determining what the maximum is for the bivariate equation along with this triangle.
00:43
So this operates just like univariate calculus, where first, in order to find the maximum, we want to find any points where the derivative equals zero.
00:56
But then we also want to investigate the points at the boundaries of our range, that are the domain that we're concerned with.
01:05
And so first, for practice, we're going to take the derivative of the equation, both with respect to x and y.
01:12
And at any point where both the derivative with respect to x and the derivative with respect to y are equal to zero, that's going to be called a stationary point, which will be get further examination as to whether that's an absolute maximum or minimum.
01:27
So first df, dx, again this operates, since we're only taking the derivative with respect to x, this is just like normal univariate calculus.
01:37
So in this case, it's going to equal 4x minus 4, and we're going to ignore all those y terms because you're not functions of x, so they become zero.
01:45
And now, with the derivative with respect to y, it's a similar process, but this one with y, so our derivative is going to be 2y minus 4.
01:54
See how these become zero and we're just left with these two terms.
01:59
Now at the point, the stationary point that we're interested in occurs when both of these derivatives are equal to zero.
02:07
So in this case, when 4x minus 4 is equal to 0, this gives x equals 1.
02:13
You can see if you plug that in, 4 minus 4 is equal to 0.
02:16
And the same process for y.
02:18
When you set this equal to 0, you're left with y equals 2, 2 times 2 minus 4 is equal to 0.
02:25
So this gives us a stationary point at 1 -2.
02:28
And you'll notice coincidentally, this is not always the case, but this is actually equal to our boundary point at 1 -2 right here.
02:36
Now we're also going to check the points 0 -2 and 0 -0 because there are also boundaries.
02:42
So the three points of interest are 1 -2, 02, and 0.
02:52
Now let's open a new page and first let's start with a point 1 -2.
02:57
So in order to test whether a stationary point is a maximum or minimum or a saddle point, we're going to take the second derivative with respect to both variables and also the derivative with respect to x and y.
03:10
So first, we're going to find the second derivative with respect to x, which if we look, our first derivative of respect to x is 4x minus 4.
03:20
And so the second derivative is going to be simply 4.
03:23
Same process with the second derivative with respect to y.
03:28
If we see it from our first derivative, this is going to be equal to 2...