Find two positive numbers satisfying the given requirements. the sum of the first and twice the second is 240 and the product is a maximum.

The two numbers are 120 and 60

Let's call the two numbers x and y. The first sentence translates into

from which we can derive

So, their product can be written as

This expression is a quadratic polynomial with negative leading coefficient, so it represents a parabola concave down. So, the vertex of the parabola is its maximum, which we can find as usual: given the parabola , its extreme point is located at .

So, in your case, since and , the maximum is located at

Now that we know y, we can deduce the value of x:

So, the two numbers are 120 and 60

- (2, -7)

- lines r and s and lines t and u must be parallel