Find the greatest possible value of the product xy, given that x and y are both positive and x + 2y = 30

The maximum value of the product x*y is 112.5

Step-by-step explanation:

We have that x + 2y = 30

so we can isolate one of the variables and get:

x = 30 - 2y

now, the product of x*y can be written as:

x*y = (30 - 2y)*y = 30y - 2y^2

now, we want to find the maximum of 30y - 2y^2

because the leading coeficient is negative (-2), we know that the hands of the quadratic function will go downwards, so the zero of the derivate of this quadratic equation will give us the value where the equation has a maximum:

f(y) = -2*y^2 + 30*y

f'(y) = -4*y + 30 = 0

y =  -30/4 = 7.5

So the maximum value is:

f(7.5) = -2*7.5^2 + 30*7.5 = 112.5

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