00:01
In this question, we are asked to find the maximum minimum values of the function f subject to the constraint xy equals to 1.
00:07
And we are going to use the lagrange multipliers method, which says that we need to solve the system of equations, the gradient of f equals to lambda times the gradient of g and xy equals to 1, where g of xy is xy minus 1.
00:32
Let's calculate the gradient of f and g.
00:35
The gradient of f is a vector which has two components, the derivative of f with respect to x, which equals to 8x, and the second component is the derivative of f with respect to y, which is 8y.
00:50
Similarly, the gradient of g is a vector which has two components.
00:56
The first component is the derivative of g with respect to x, which is y.
01:01
The second component is the derivative of g with respect to y, which is x.
01:07
The system of equations becomes 8x, 8y equals to lambda multiplied by y x and xy equals to 1.
01:23
Now let's rewrite the vector equation as two scalar equations.
01:28
The first equation is going to be 8x equals to lambda y.
01:34
The second equation is going to be 8 y equals to lambda x.
01:43
And the last equation is xy equals to 1.
01:49
Now from the first equation x equals to lambda y over 8.
01:57
Now we are going to plug in this x in the second equation.
02:00
To get 8y equals to lambda multiplied by lambda y over 8.
02:13
And the last equation is going to be same.
02:18
Now in the second equation we are going to get that 64y.
02:24
So we are going to rewrite the first equation.
02:29
The second equation becomes 64y equals to lambda squared y.
02:38
And xy equals to 1.
02:47
Now note that y cannot be equal to 0, because if y equals 0, in the last equation we are going to get 0 equals to 1.
02:55
So y is not 0, which means we can cancel y in the second equation.
03:01
And we are going to get that lambda squared equals to 64 or lambda equals to plus minus 8.
03:18
So when lambda equals to 8, we are going to get that x equals 8 y over 8, which equals to y over 8, and then if x, replacing x by y in the last equation, we are going to get that y squared equals to 1.
03:41
That's for lambda equals to 8.
03:44
For lambda equals to negative 8, we are going to get that x equals to negative y.
03:54
And if we replace x by negative y in the last equation, we are going to get that negative y squared equals to 1.
04:07
Or, y squared equals to negative 1, which is impossible because a square of a number cannot be negative.
04:20
Let me rewrite this.
04:30
Squared equals to negative 1 is impossible...