Question

Find two positive numbers satisfying the given requirements. The product is 192 and the sum of the first plus three times the second is a minimum.

Short answer

The optimal values for the numbers are given by the solution to the system of equations, found from the derivative and the product equation.

Step-by-step solution

Set Up the Equations

Given that the product of two numbers \(x\) and \(y\) should be 192, this translates to \(xy = 192\). It is also stated that the sum of the first number (x) and three times the second number (y) i.e. \(x + 3y\) should be at a minimum. Hence, the objective is to minimize \(x + 3y\).

Express x in terms of y from first equation

We can express \(x\) in terms of \(y\): \(x=192/y\). This is derived from the first equation given in the problem.

Substitute x in the minimized equation

In order to minimize the second equation, we need to have it in terms of one variable. Hence let's substitute the value of \(x\) obtained from Step 2 into the equation to be minimized. The equation to minimize becomes \(192/y + 3y\).

Derive the minimization function

The problem deals with the minimization of \(192/y + 3y\). To find the minimum point, compute the derivative of this function. The derivative helps us to locate the points on the function where there might be a maximum or a minimum. So find the derivative of \(f(y) = 192/y + 3y\) with respect to \(y\). The result is \(f'(y)= -192/y^2 +3\).

Find the critical points

Critical points are those where the derivative of the function is either zero or does not exist. Set \(f'(y)= -192/y^2 +3 = 0\), and solve for \(y\). Alternatively, if the equation cannot be solved, find points where the derivative does not exist. These are possible locations for a minimum value.

Find the minimum

Classify the critical points. Typically, this is done by using the second derivative test, or by considering the values around the critical points. Substitute these points into the original function to find the minimal value.

Find the other number

Substitute the value of \(y\) into the equation \(x = 192/y\) to find the value of \(x\). The pair \((x,y)\) will be the two numbers which satisfy the given conditions.