Minimize a Product of Two Numbers with Given Difference
If you find this page helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor
The difference of two numbers is 20. Determine such a pair of numbers which has a product as small as possible.Let's choose variables a and b to represent the numbers.
What other variables do we need?
We need the difference of the numbers and their product
Let D be the difference and P be their product.
Express the difference 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 a.
Then we can differentiate to find the value of a for a minimum product. This value of a can be used in equation 1 to determine b.
Solve equation 1 for b.
Equation 3
Substitute equation 3 into equation 2.
We get
Equation 4
Differentiate.
Equation 5
How do we determine the value of a which minimizes the product?
We set this result to 0 and solve.
Do that.
Determine the corresponding value of b from equation 3.
So the two numbers, 10 and –10, result in the minimum product.
Determine this minimum product from equation 2.
How can we check that this is a minimum by substituting values?
We can try other numbers which satisfy equation 1.
One such pair, close to the pair we have found, is 11 and –9. Find their product.
Their product is –99.
Is this larger than –100, which we think is a minimum?
Yes. –99 > –100. Recall that with negative numbers, greater means "less negative".
Apparently we do have a minimum. How can we check via differentiation?
We can take the second derivative of P with respect to a and check the algebraic sign when
Determine the second derivative using equation 5.
Determine the sign of the second derivative when a = 10.
Since the second derivative is +2, independent of a, the sign is positive.
Is this consistent with a minimum?
Yes, the change in the first derivative is to become more positive at a minimum.
Let's graph equation 4 for 0 < x < 25 for further confirmation. Try to do this and then check with the graph below.
Graph of the Product versus the larger 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