0:00
All right.
00:01
So here we want to find two positive real numbers such that.
00:05
So we have one constraint.
00:07
So something has to be such that x plus y, two numbers x plus y, such that they add up to 105.
00:17
But such that when we take the product of the first and we square the second, we maximize this function.
00:25
Right.
00:26
So this is of course a maximization with equality constraint problems.
00:29
And we're going to do.
00:30
Use the method of lagrange multipliers, right? so the first step is to define our lagrangian function, l of x, y, y, and a lagrange multiplier lambda.
00:41
And this is our function, x times y squared, minus lambda or plus lambda, it doesn't matter really, 105 minus x minus y.
00:51
So we want to find the maximum or the minimum of this function.
00:55
And you know how the method goes, right? we first calculate all the partial derivatives, the l, dx, equals.
01:01
The l d y equals 0 and d l d lambda equals 0 so let's go about solving this where d l d x is y squared from here and from here we get a plus lambda is equal to zero from here for y we get two times x times y and also a plus lambda is equal to zero and in the last one we just get x plus y equals to 105 so let's see what we can conclude from here, right? so the first equation, for example, gives lambda equals minus y squared, which you can replace on the second one to obtain 2xy minus y squared equals to 0.
01:46
And here's just a constraint.
01:47
I'm not going to repeat it.
01:50
Now let's use the second equation to get some conclusions...