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Introduction

In mathematics, we often encounter problems that require us to find the maximum or minimum value of a function. One such problem is to find the pair of positive numbers whose sum is a given value and whose product is the largest possible. In this article, we will explore this problem and find the pair of numbers that satisfies the given conditions.

Problem Statement

We are given a pair of positive numbers whose sum is 15. We need to find the pair of numbers that will produce the largest possible product.

Understanding the Problem

To solve this problem, we need to understand the relationship between the sum and product of two numbers. Let's consider two positive numbers, x and y, whose sum is 15. We can write this as an equation:

x + y = 15

We want to find the pair of numbers that will produce the largest possible product, which is given by:

P = xy

Analyzing the Problem

To maximize the product P, we need to find the values of x and y that will give us the maximum value of P. Since the sum of x and y is fixed at 15, we can express y in terms of x as:

y = 15 - x

Substituting this expression for y into the equation for P, we get:

P = x(15 - x)

Expanding this equation, we get:

P = 15x - x^2

Finding the Maximum Product

To find the maximum value of P, we need to find the value of x that will give us the maximum value of P. We can do this by taking the derivative of P with respect to x and setting it equal to zero:

dP/dx = 15 - 2x = 0

Solving for x, we get:

x = 7.5

Substituting this value of x into the equation for y, we get:

y = 15 - 7.5 = 7.5

Conclusion

Therefore, the pair of numbers that will produce the largest possible product is (7.5, 7.5). This makes sense, since the product of two numbers is maximized when the numbers are equal.

Discussion

This problem is a classic example of a maximization problem in mathematics. It requires us to find the values of x and y that will give us the maximum value of P, subject to the constraint that the sum of x and y is 15. We can solve this problem using calculus, by taking the derivative of P with respect to x and setting it equal to zero.

Example

Let's consider an example to illustrate this problem. Suppose we have two numbers, 5 and 10, whose sum is 15. We can calculate their product as:

P = 5(10) = 50

However, this is not the maximum possible product. To find the maximum product, we need to find the pair of numbers that will give us the maximum value of P. Using the method described above, we can find that the pair of numbers that will produce the largest possible product is (7.5, 7.5).

Applications

This problem has many applications in real-world scenarios. For example, in economics, we may want to find the pair of prices that will maximize the revenue of a company, subject to a given constraint on the total cost. In engineering, we may want to find the pair of dimensions that will maximize the volume of a container, subject to a given constraint on the surface area.

Conclusion

In conclusion, the problem of finding the pair of numbers that will produce the largest possible product is a classic example of a maximization problem in mathematics. We can solve this problem using calculus, by taking the derivative of P with respect to x and setting it equal to zero. The pair of numbers that will produce the largest possible product is (7.5, 7.5). This problem has many applications in real-world scenarios, and is an important tool for solving optimization problems.

References

  • [1] "Calculus" by Michael Spivak
  • [2] "Mathematics for Economists" by Carl P. Simon and Lawrence Blume
  • [3] "Optimization Techniques" by S. S. Rao

Further Reading

For further reading on this topic, we recommend the following resources:

  • [1] "Calculus" by Michael Spivak
  • [2] "Mathematics for Economists" by Carl P. Simon and Lawrence Blume
  • [3] "Optimization Techniques" by S. S. Rao

Glossary

  • Maximization problem: A problem that requires us to find the maximum value of a function, subject to a given constraint.
  • Calculus: A branch of mathematics that deals with the study of rates of change and accumulation.
  • Derivative: A measure of how a function changes as its input changes.
  • Optimization problem: A problem that requires us to find the best solution, subject to a given constraint.
    Q&A: Maximizing the Product of Two Numbers with a Given Sum ===========================================================

Q: What is the problem of maximizing the product of two numbers with a given sum?

A: The problem of maximizing the product of two numbers with a given sum is a classic example of a maximization problem in mathematics. We are given a pair of positive numbers whose sum is a given value, and we need to find the pair of numbers that will produce the largest possible product.

Q: How do we solve this problem?

A: To solve this problem, we can use calculus to find the maximum value of the product function. We can take the derivative of the product function with respect to one of the variables, set it equal to zero, and solve for the variable.

Q: What is the formula for the product function?

A: The formula for the product function is P = xy, where x and y are the two numbers whose sum is a given value.

Q: How do we find the maximum value of the product function?

A: To find the maximum value of the product function, we can take the derivative of the product function with respect to one of the variables, set it equal to zero, and solve for the variable. This will give us the critical point(s) of the function, which may correspond to the maximum value.

Q: What is the critical point of the product function?

A: The critical point of the product function is the value of the variable that makes the derivative of the function equal to zero. In this case, the critical point is x = 7.5.

Q: What is the maximum value of the product function?

A: The maximum value of the product function is the value of the function at the critical point. In this case, the maximum value is P = 7.5(7.5) = 56.25.

Q: What are some real-world applications of this problem?

A: This problem has many real-world applications, such as:

  • Finding the pair of prices that will maximize the revenue of a company, subject to a given constraint on the total cost.
  • Finding the pair of dimensions that will maximize the volume of a container, subject to a given constraint on the surface area.
  • Finding the pair of inputs that will maximize the output of a production process, subject to a given constraint on the resources available.

Q: How do we use calculus to solve this problem?

A: To use calculus to solve this problem, we can take the derivative of the product function with respect to one of the variables, set it equal to zero, and solve for the variable. This will give us the critical point(s) of the function, which may correspond to the maximum value.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

  • Not taking the derivative of the product function correctly.
  • Not setting the derivative equal to zero correctly.
  • Not solving for the variable correctly.
  • Not checking the second derivative to ensure that the critical point corresponds to a maximum value.

Q: How do we check if the critical point corresponds to a maximum value?

A: To check if the critical point corresponds to a maximum value, we can take the second derivative of the product function and evaluate it at the critical point. If the second derivative is negative, then the critical point corresponds to a maximum value.

Q: What are some advanced topics related to this problem?

A: Some advanced topics related to this problem include:

  • Using Lagrange multipliers to solve constrained optimization problems.
  • Using the method of steepest descent to solve optimization problems.
  • Using the method of gradient descent to solve optimization problems.
  • Using the method of Newton's method to solve optimization problems.