00:01
Okay, so in this exercise we have our function f of x, y, z equal to 4y cubed sine of x e to the 3yz.
00:14
Perfect.
00:15
We have our point p equal to 0, 1, 2 and we need to find the direction of the maximum increase of f at p.
00:25
Well, what is this direction? this direction is clearly going to be the gradient of f evaluated at p over its length.
00:40
Perfect.
00:41
So let's find the gradient of f.
00:44
Well, this is pretty easy.
00:47
The gradient of f, the first coordinate of the gradient is the partial derivative of this guy with respect to x.
00:54
So, 4y cubed cosine of x e to the 3yz.
01:04
The second coordinate, well, the second coordinate is the partial derivative of this guy with respect to y.
01:12
So we are going to have sine of x multiplied by some function depending on y and z only.
01:20
And we are not going to compute this function.
01:23
The reason is going to be clear in one moment.
01:27
So let me write sine of x multiplied by g of y, z.
01:32
Similarly, the third coordinate is going to be sine of x multiplied by a function depending on y and z only.
01:42
And we are not going to compute this function because, as we can see, sine of 0 is 0.
01:51
So, the gradient of f evaluated at p, well, the gradient of f evaluated at p is going to have the first coordinate different from 0 and the other two coordinates equal to 0...