Exercise · ZW-LAGRANGE-MULTIPLIERS-004 · ★☆☆ Basic
Find the maximum value of \(f(x,y)=xy\) on the line \(2x+y=12\).
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- D. 18
Explanation
From \(y=2\lambda\) and \(x=\lambda\) we get \(y=2x\); with \(2x+y=12\) this gives \(x=3\), \(y=6\) and \(f=18\).
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-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-006 Find the maximum value of f(x,y)=xy on the circle x^2+y^2=8.
- 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?