Maximize the Product of Two Numbers: Example 1
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The sum of one number and five times another number is 50.Determine such a pair of numbers which has a product as large as possible.
Let's choose variables a and b to represent the numbers.
What other variables do we need?
We need the sum of the numbers and their product
Let S be the sum condition and P be their product.
Express the sum condition in terms of a and b.
Equation 1
Express the product in terms of a and b.
Equation 2
How can we find a and b?
We have two equations and two unknowns.
Hence we can combine the equations to express the product in terms of one variable, say b.
Then we can differentiate to find the value of b for a maximum product. This value of b can be used in equation 1 to determine the corresponding value of a.
Solve equation 1 for a.
Equation 3
Substitute equation 3 into equation 2.
We get
Equation 4
Differentiate.
Equation 5
How do we determine the value of b which maximizes the product?
We set this result to 0 and solve.
Do that.
Determine the corresponding value of a from equation 3.
So the two numbers, 25 and 5, result in the maximum product.
Determine this maximum product from equation 2.
How can we check that this is a maximum by substituting values?
We can try other numbers which satisfy equation 1.
One such pair, close to the pair we have found, is 20 and 6. Find their product.
Their product is 120, less than our maximum of 125. For the pair 30 and 4, we also get a product of 120.
Apparently we do have a maximum. How can we check via differentiation?
We can take the second derivative of P with respect to b and check the algebraic sign when
Determine the second derivative using equation 5.
Since the second derivative is –10, independent of b, the sign is negative at b = 5.
Is this consistent with a maximum?
Yes, the change in the first derivative is to become more negative at a maximum.
Let's graph equation 4 for 0 < x < 10 for further confirmation. Try to do this and then check with the graph below.
graph of the product versus the smaller of the two numbers
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor