Can the Inequality x^x + y^y < (x+y)^(x+y) be Proven Algebraically?

  • Context: Graduate 
  • Thread starter Thread starter bill01
  • Start date Start date
  • Tags Tags
    Inequality
Click For Summary
SUMMARY

The inequality x^x + y^y < (x+y)^(x+y) holds true for all x, y ≥ 1. Participants in the discussion suggest using derivatives to analyze the slope changes of both sides of the inequality. The expansion of (x+y)^(x+y) demonstrates that it is greater than x^x and y^y, as all terms in the expansion are positive when x and y are greater than 1. An algebraic proof is preferred over graphical methods for a definitive solution.

PREREQUISITES
  • Understanding of algebraic inequalities
  • Knowledge of calculus, specifically derivatives
  • Familiarity with exponential functions
  • Ability to perform polynomial expansions
NEXT STEPS
  • Study algebraic proofs of inequalities
  • Learn about derivatives and their applications in inequality analysis
  • Explore polynomial expansion techniques
  • Investigate the properties of exponential functions in mathematical proofs
USEFUL FOR

Mathematicians, students studying calculus and algebra, and anyone interested in proving mathematical inequalities.

bill01
Is it possible to prove this:
x^x + y^y < (x+y)^(x+y) for every x,y >=1 ?
 
Mathematics news on Phys.org
Well, let me think, since I am not really sure if this would work for all parameters of x and y... never mind, you have x,y >1! What you could do is take the derivative of both equations to measure it's change in slope, and if (x+y)^(x+y)is greater, then it will have a change in slope that is greater then the other equation. But I am not sure if that is what you want.
 
Iam just proving it,just expand the Right side (x+y)^(x+y), u get x multiplied by x+y times which is obviously greater than x^x and same in case of y and remaining terms of expansion are positive as x,y>1 and no negative terms in expansion. hope it helps.
 
  • Like
Likes   Reactions: RaulTheUCSCSlug
Thanks for the answers, but I would prefer an algebraic solution.
I did what Raul said with the graph but I would like an algebraic sol.
I believe that it is solved algebraically and it is not a transcendental equation.
 
Last edited by a moderator:

Similar threads

High School Inverse Functions: Why rewrite as y=f(x) ?
  • · Replies 11 ·
Replies
11
Views
2K
MHB Solve Inequality & Simul Eqns: Alg Workings Q (a,b,c)
  • · Replies 1 ·
Replies
1
Views
1K
MHB Prove Divisibility: $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$
  • · Replies 1 ·
Replies
1
Views
2K
High School Have I proved some part of Fermat's last theorem?
  • · Replies 15 ·
Replies
15
Views
2K
MHB Show x³y+y³z+z³x is a constant
  • · Replies 1 ·
Replies
1
Views
1K
MHB Solving $x^3+y^3=7$ and $x^2+y^2+x+y+xy=4$ System of Equations
  • · Replies 1 ·
Replies
1
Views
1K
MHB Maximize Σ(x^2+y)(y^2+x))/((x+y-1)^2)
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Can you solve unknown triangle from shared hypotenuse?
  • · Replies 7 ·
Replies
7
Views
2K
MHB Solving System of Equalities: x^2-y\sqrt xy & y^2-x\sqrt xy =3
  • · Replies 2 ·
Replies
2
Views
1K
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