00:01
There are two positive numbers x and y such that x plus y equals 50.
00:05
Find the maximum of 2x plus 3y squared.
00:10
So we know that x and y both must be positive so x must be greater than zero and y must be greater than zero.
00:21
So we have the quantity 2x plus 3y squared we want to maximize.
00:26
So first let's see if we can get this in terms of just x.
00:28
So we know that x plus y equals 50.
00:32
We can easily solve this for y.
00:35
Y would be 50 minus x.
00:38
Now we can write this expression as 2x plus 3 and we can substitute 50 minus x for y.
00:46
So we have 50 minus x squared.
00:51
And then we could easily square this binomial here.
00:55
50 squared is 2500 and then we would have minus 100x for our middle term and then plus x squared.
01:06
And then we would have 2x plus we distribute the 3 that gives us 7500 minus 300x plus 3x squared.
01:17
And if we get that in standard form that would just be 3x squared minus 298x plus 7500.
01:28
So we can write that as a function in terms of x.
01:32
So we do need to get some restrictions on x.
01:35
We already know that x must be positive since x must be greater than zero.
01:41
And we also know that y which we said y was equal to 50 minus x.
01:49
Y must also be greater than zero.
01:53
So that means 50 minus x must be greater than zero since 50 minus x is equal to y.
02:01
So if we solve this inequality for x we add x to both sides.
02:05
That gives 50 greater than x or we can rewrite that as x is less than 50.
02:12
So that gives the domain of this function that x must be less than 50 but still greater than zero...