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Lagrange multipliers practice problems

Maximize or minimize subject to a constraint: ∇f = λ∇g. 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

3 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 10.55

medium Video
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
  1. Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(4 x + y\right)\\\frac{\partial}{\partial y} \left(4 x + y\right)\end{matrix}\right] = \left[\begin{matrix}4\\1\end{matrix}\right] \]
    ∇f.✓ Proved
  3. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 25\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 25\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]
    ∇g.✓ Proved
  4. So x = 4/(2λ), y = 1/(2λ); substituting into the constraint gives λ = ±√(17)/(2√25).
    Reviewed
  5. \[ 5 \sqrt{17} \]
    The maximum; the minimum is its negative.Reviewed
Answer \( \max = 5 \sqrt{17},\ \min = - 5 \sqrt{17} \)
A contour map of f(x, y) = 4x + y: each curve joins points where f has the same value. The constraint circle of radius 5 is drawn over it; the largest and smallest values on the circle are where a level line just touches it.

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.60

medium Video
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
  1. Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x + 3 y\right)\\\frac{\partial}{\partial y} \left(2 x + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}2\\3\end{matrix}\right] \]
    ∇f.✓ Proved
  3. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 4\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 4\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]
    ∇g.✓ Proved
  4. So x = 2/(2λ), y = 3/(2λ); substituting into the constraint gives λ = ±√(13)/(2√4).
    Reviewed
  5. \[ 2 \sqrt{13} \]
    The maximum; the minimum is its negative.Reviewed
Answer \( \max = 2 \sqrt{13},\ \min = - 2 \sqrt{13} \)
A contour map of f(x, y) = 2x + 3y: each curve joins points where f has the same value. The constraint circle of radius 2 is drawn over it; the largest and smallest values on the circle are where a level line just touches it.

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.197

medium Video
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
  1. Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x + 4 y\right)\\\frac{\partial}{\partial y} \left(x + 4 y\right)\end{matrix}\right] = \left[\begin{matrix}1\\4\end{matrix}\right] \]
    ∇f.✓ Proved
  3. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 9\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 9\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]
    ∇g.✓ Proved
  4. So x = 1/(2λ), y = 4/(2λ); substituting into the constraint gives λ = ±√(17)/(2√9).
    Reviewed
  5. \[ 3 \sqrt{17} \]
    The maximum; the minimum is its negative.Reviewed
Answer \( \max = 3 \sqrt{17},\ \min = - 3 \sqrt{17} \)
A contour map of f(x, y) = x + 4y: each curve joins points where f has the same value. The constraint circle of radius 3 is drawn over it; the largest and smallest values on the circle are where a level line just touches it.

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More lagrange multipliers practice

Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.54medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.56medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.57medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.58medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.59medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.61medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.62medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.63medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.64medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.65medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.66medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.67medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.68medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.108medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.109medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.110medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.111medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.112medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.113medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.114medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.115medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.116medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.117medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.118medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.119medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.120medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.121medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.122medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.162medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.163medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.164medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.165medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.166medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.167medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.168medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.169medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.195medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.196medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.198medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.199medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.200medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.201medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.202medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.227medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.228medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.229medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.230medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.231medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.232medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.233medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.234medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.259medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.260medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.261medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.262medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.263medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.264medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.265medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.266medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.291medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).
Problem 10.292medium Video✓ Nihil obstat
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.293medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.294medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.295medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.296medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.297medium Video✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.298medium Video✓ Every equation proved