Exercise 1597Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

Lagrange with exponential objective

#Lagrange#exponential#constrained maximum

Statement

Maximize $f(x,y)=e^{xy}$ subject to $x+y=1$ .

Hint

Maximize $xy$ under $x+y=1$ via AM-GM.

Show solutionHide solution

AM-GM. $xy\le\left(\frac{x+y}{2}\right)^2=\frac14$ , with equality iff $x=y=1/2$ .

Conclusion. Maximum $e^{1/4}$ at $(1/2,1/2)$ .