Step 1 :First, we need to define the variables and functions. We will use the symbols function to define the variables \(x\), \(y\), and \(\lambda\), and the function \(L\) for the Lagrangian.
Step 2 :We will also compute the derivatives of \(L\) with respect to \(x\), \(y\), and \(\lambda\).
Step 3 :Finally, we will solve the system of equations.
Step 4 :The solution to the system of equations is \(x = 42540\), \(y = 42540\), and \(\lambda = 1.31578947368421e-6\).
Step 5 :This means that the company should spend \$42540 on development to maximize the number of products sold.
Step 6 :Final Answer: The company should spend \(\boxed{\$ 42540}\) on development to maximize the number of products sold.