00:01
So in this question, they say among all the pairs of numbers whose sum is 16, i want to find a pair whose product is as large as possible.
00:12
I want to know what is that maximum product.
00:16
Does this function have a maximum value or the minimum value? explain.
00:21
What is the domain of our function? what is the range of our function? so they tell us i have two numbers.
00:32
Pairs of numbers whose sum is 16.
00:37
So they are telling me that x plus y is equal to 16.
00:44
And i am trying to find a pair whose product is as large as possible, so i am trying to maximize the function p which is equal to x times y.
01:00
Now that's currently a function of two variables, but i want to squash this down to the world of one variable.
01:08
To do that, i'm going to go to my constraint, x plus y equals 16, and i'm going to solve for one of the variables.
01:17
I am going to note that my y is equal to 16 minus x this time.
01:24
I'm going to take that, and i'm going to plug it into my function p.
01:28
So my p is equal to x times the quantity of, 16 minus x this time.
01:37
And if i distribute, this means that my p itself would be equal to 16x minus x squared.
01:49
Now, i'm going to take the derivative...