Exercise 862Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

Maximizing a product

#Lagrange#product#constraint

Statement

Maximize $f(x,y)=(xy)^a$ for $x,y\ge0$ , $a>0$ , under $2x+3y=12$ .

Hint

Lagrange on the interior $x,y>0$ ; check the boundary separately.

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Lagrange. $a x^{a-1}y^a=2\lambda$ , $a x^ay^{a-1}=3\lambda$ . Ratio: $y/x=2/3$ , i.e. $3y=2x$ . Constraint: $2x+2x=12$ , $x=3$ , $y=2$ .

Boundary. On $x=0$ or $y=0$ , $f=0$ . At $(3,2)$ , $f=6^a>0$ .

Conclusion. Global maximum $6^a$ at $(3,2)$ .