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Lagrange multipliers practice problems

Maximize or minimize subject to a constraint: ∇f = λ∇g. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.

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✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
Problem 10.55medium✓ 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✓ 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✓ 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✓ 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✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.60medium✓ 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✓ 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✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
Problem 10.63medium✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.64medium✓ 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✓ 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✓ 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✓ 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✓ 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✓ 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✓ 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✓ 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✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.112medium✓ 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✓ 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✓ 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✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
Problem 10.116medium✓ 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✓ 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✓ 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✓ 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✓ 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✓ Every equation proved
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
Problem 10.122medium✓ Every equation proved