Maximizing xy: Understanding Optimization Problems in Mathematics

  • Context: Undergrad 
  • Thread starter Thread starter Mr Davis 97
  • Start date Start date
  • Tags Tags
    Optimization
Click For Summary
SUMMARY

The discussion focuses on optimization problems in mathematics, specifically maximizing the product of two variables, x and y, under the constraint x + y = 2. The conclusion drawn is that the maximum value of the product xy is 1, achieved when x equals y. This is supported by the Arithmetic Mean-Geometric Mean (AM-GM) inequality, which establishes that while 2 is an upper bound, 1 is the least upper bound and thus the maximum value. The distinction between upper bound and least upper bound is crucial for understanding optimization in this context.

PREREQUISITES
  • Understanding of optimization problems in mathematics
  • Familiarity with the Arithmetic Mean-Geometric Mean (AM-GM) inequality
  • Basic algebraic manipulation and equations
  • Knowledge of upper bounds and least upper bounds in mathematical analysis
NEXT STEPS
  • Study the applications of the AM-GM inequality in various optimization problems
  • Explore advanced topics in mathematical optimization techniques
  • Learn about constraints in optimization and how they affect solutions
  • Investigate the differences between upper bounds and least upper bounds in mathematical contexts
USEFUL FOR

Students and professionals in mathematics, particularly those studying optimization problems, as well as educators seeking to clarify concepts related to bounds and inequalities.

Mr Davis 97
Messages
1,461
Reaction score
44
I am little confused when it comes to optimization problems. For example, say we are given that ##x+y=2##, and are asked to maximize ##xy##. By AM-GM, we have that ##xy \le 1##. But why should this indicate that ##1## is the maximum value? Isn't it an equally true statement to claim that ##xy \le 2##, since the former interval is contained in the latter?
 
Physics news on Phys.org
It is equally true, but not as useful. 2 is merely an upper bound, whereas 1 is a least upper bound. In fact it is a maximum, that is achieved when ##x=y##. Arithmetic and Geometric Means are identical when all data are the same.
 
  • Like
Likes   Reactions: Mr Davis 97
andrewkirk said:
It is equally true, but not as useful. 2 is merely an upper bound, whereas 1 is a least upper bound. In fact it is a maximum, that is achieved when ##x=y##. Arithmetic and Geometric Means are identical when all data are the same.
I think it was the distinction between upper bound and least upper bound that I was looking for.
 

Similar threads

Undergrad Understanding Mixed Partial Derivatives: How Do You Solve Them?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Problem with function approximation
  • · Replies 12 ·
Replies
12
Views
1K
Undergrad Multivariable optimization problem
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Is Being Path-Connected the Key to Understanding Simply Connected Regions?
  • · Replies 6 ·
Replies
6
Views
2K
DP: proving existence of optimal substructure for "Sherlock and Cost"
  • · Replies 17 ·
Replies
17
Views
2K
Graduate Constructing PDFs for Max Likelihood Density Estimation Problem
  • · Replies 2 ·
Replies
2
Views
2K
MHB Solve Optimization Problem: Find Min & Max of f(x,y)
  • · Replies 3 ·
Replies
3
Views
2K
Maximize xy when x + y + z = 3
  • · Replies 2 ·
Replies
2
Views
2K
Placing a pole with maximal radius subject to constraints
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Lagrange Multiplier. Dealing with f(x,y) =xy^2
  • · Replies 1 ·
Replies
1
Views
1K