MHB Absolute value of real numbers

Click For Summary
SUMMARY

The discussion centers on the optimization problem involving real numbers \(x\), \(y\), and \(z\) that satisfy the equation \(3x + 2y + z = 1\). The goal is to maximize the expression \(\dfrac{1}{1+|x|}+\dfrac{1}{1+|y|}+\dfrac{1}{1+|z|}\) under the constraint provided. The maximum value is expressed as \(\dfrac{q}{p}\), where \(p\) and \(q\) are relatively prime positive integers. The final solution yields \(p + q = 5\).

PREREQUISITES
  • Understanding of linear equations and inequalities
  • Familiarity with optimization techniques in calculus
  • Knowledge of absolute value functions
  • Basic concepts of number theory, particularly relative primality
NEXT STEPS
  • Study optimization methods in calculus, focusing on constrained optimization
  • Explore the properties of absolute value functions and their graphical representations
  • Learn about linear programming and its applications
  • Investigate number theory concepts, specifically relative primality and its implications
USEFUL FOR

Mathematicians, students studying calculus and optimization, and anyone interested in number theory and real analysis.

anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Reals $x,\,y$ and $z$ satisfies $3x+2y+z=1$. For relatively prime positive integers $p$ and $q$, let the maximum of $\dfrac{1}{1+|x|}+\dfrac{1}{1+|y|}+\dfrac{1}{1+|z|}$ be $\dfrac{q}{p}$. Find $p+q$.
 
Mathematics news on Phys.org
12.png
 

Similar threads

Undergrad ##x\lt y \lt z## positive integer numbers and if ##\dfrac 1x+\dfrac 1y+\dfrac 1z=\dfrac 13##
  • · Replies 6 ·
Replies
6
Views
2K
MHB Inequality involving positive real numbers
  • · Replies 1 ·
Replies
1
Views
1K
MHB Roots of Polynomial: Find $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Real Number Pairs $(p,\,q)$ Satisfying Inequality
  • · Replies 2 ·
Replies
2
Views
1K
MHB What is the remainder when m+n is divided by 1000 in a trigonometric challenge?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Finding Real Part of $z$ for Complex Numbers
  • · Replies 1 ·
Replies
1
Views
1K
MHB Finding Min Value of $\dfrac{|b|+|c|}{a}$ from Roots of Cubic Equations
  • · Replies 4 ·
Replies
4
Views
1K
MHB Real Numbers $a,\,b,\,c$ Solving System of Equations
  • · Replies 2 ·
Replies
2
Views
1K
MHB Real Roots of Composite Polynomials: Solving P(Q(x))=0
  • · Replies 1 ·
Replies
1
Views
1K
MHB Min Value of $\dfrac{a+3c}{a+2b+c}$+$\dfrac{4b}{a+b+2c}$+$\dfrac{8c}{a+b+3c}$
  • · Replies 1 ·
Replies
1
Views
2K