Exercise · ZW-LAGRANGE-MULTIPLIERS-006 · ★★☆ Intermediate
Find the maximum value of \(f(x,y)=xy\) on the circle \(x^2+y^2=8\).
Choose one option.
Show answer
- A. 4
Explanation
The equations \(y=2\lambda x\), \(x=2\lambda y\) give \(x^2=y^2\), so \(x=\pm y\) and \(2x^2=8\). The points \((2,2)\) and \((-2,-2)\) give \(4\); the other two give \(-4\).
Review the concept: Lagrange multipliers
The method of Lagrange multipliers finds the extrema of a function \(f\) on the curve \(g=k\) among the points where the gradient of \(f\) is a multiple \(\lambda\) of the gradient of \(g\).
\[\nabla f=\lambda\,\nabla g,\qquad g=k\]
More practice problems: lagrange multipliers
- ZW-LAGRANGE-MULTIPLIERS-001 Find the maximum value of f(x,y)=x+y on the circle x^2+y^2=2.
- ZW-LAGRANGE-MULTIPLIERS-002 Find the minimum value of f(x,y)=x^2+y^2 on the line x+2y=5.
- ZW-LAGRANGE-MULTIPLIERS-003 To maximize f(x,y)=xy subject to x+y=6, which system does the method set up?
- ZW-LAGRANGE-MULTIPLIERS-004 Find the maximum value of f(x,y)=xy on the line 2x+y=12.
- ZW-LAGRANGE-MULTIPLIERS-005 When minimizing f(x,y)=x^2+y^2 subject to x+y=4, what is λ at the optimal point?
- ZW-LAGRANGE-MULTIPLIERS-007 Find the minimum value of f(x,y)=x+2y on the circle x^2+y^2=5.
- ZW-LAGRANGE-MULTIPLIERS-008 Find the minimum value of f(x,y,z)=x^2+y^2+z^2 on the plane x+y+z=3.
- ZW-LAGRANGE-MULTIPLIERS-009 What does ∇ f=λ∇ g mean geometrically at an extremum of f on the curve g=k?