00:01
In this question, we are asked to find the maximum and minimum values of a function of two variables subject to the constraints x squared over 16 plus y squared over 81 equals 1.
00:16
And to do that, we are going to use the method of lagrange multipliers.
00:22
So recall that by that method, we need to create a function g, which is basically the constraint condition, with one move to the left hand side.
00:42
And note that you remove the equation condition, you just move one to the left hand side and you call the left hand side by a new function g, right? it doesn't have to be equal to 0.
00:54
Now we are going to use lagrange multipliers.
00:59
By the lagrange multipliers, we want the gradient of the function f to be a multiple of the gradient of the function g.
01:10
Here lambda is a scaling coefficient right so and also we have the constraint condition right constraint condition is this equation here x squared over 16 plus y squared over 81 equals 1 and this gives us a system of linear equations now let's calculate the gradient of f the gradient of of f, recall that by definition it's a vector fx, fy.
01:56
So the derivative of f with respect to x is 9y, and the derivative of f with respect to y is 9x.
02:07
Now let's calculate the gradient of g.
02:13
It's gx, g, g, x, comma, gy.
02:18
The derivative of g with respect to x is 2x over 16, which is same as x over 8.
02:27
So it's 2x over 16.
02:28
And the gradient of g with respect the derivative of g with respect to y is 2y over 81 so we're going to get x over 16 and 2 y over 81 now we want them to be equal right so we want the gradient of f to be equal to lambda times the gradient of g and of course we have the second condition x squared over over 16 plus y squared over 81 equals 1 now this gives up we can rewrite this as three equations 9y equals lambda times x over 16 9x equals lambda times 2x over sorry 2y over 81 and the last equation this is a system of equations now let's try to solve the system of equations so let's let's do, let's find lambda from the second equation.
04:08
From the second equation, lambda equals to 16 times 9y over x.
04:18
From the first equation, from the second equation, lambda equals to 9 times 81 x over 2y.
04:29
And the last equation is still same.
04:39
Now the first two equations, the right hand sides in the first two equations are equal to lambda, therefore they're equal to each other.
04:49
So we're going to write it as lambda equals to 16 times 9 y over x.
05:00
Now we're going to plug in 16 times 9 y over x in the second equation for lambda...