Question

Find two positive numbers satisfying the given requirements. The sum of the first and twice the second is 100 and the product is a maximum.

Short answer

The two positive numbers satisfying the given requirements are 50 and 25.

Step-by-step solution

Express one variable using the constraint equation

Let's denote the two numbers as \(x\) and \(y\). From the problem, we have \(x + 2y = 100\). This equation can be rearranged to express \(x\) in terms of \(y\). So, we can express \(x\) as \(x = 100 - 2y\).

Formulate the objective function

We are asked to maximize the product of the two numbers. The product \(xy\) can be expressed as a function of \(y\) using the substitution \(x = 100 - 2y\). After this substitution we obtain the product function \(P(y)=y(100-2y)= 100y-2y^2\).

Differentiate the product function

To set up for finding the maximum value, we need to find the derivative of the resulting function to set up for finding its critical points. The derivative of our function \(P(y)\) is \(P'(y)=100-4y\).

Compute the maximum value

We set the derivative equal to zero and solve for \(y\): \(100 - 4y = 0\). Solving for \(y\) we find \(y = 25\).

Determine the other variable

Substitute \(y = 25\) into the equation from the first step (\(x = 100 - 2y\)) to get \(x = 100 - 2*25 = 50\).

Verification

To confirm if the obtained values are indeed giving the maximum product, one can substitute near values to \(y\) into the function \(P(y)\) and confirm that for no other \(y\) value, \(P(y)\) yields a higher value.