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(1) Lagrange multiplier is a very useful technique to determine the extremum of a function under constraint(s). For a function f(x, y), the extremum (minimum or maximum) is determined by evaluating of /ax = af/ay = 0. But determining the extremum under a constraint g(x, y) = 0 can be tricky. Even if y can be expressed explicitly in terms of x from g(x, y) = 0, the calculations may become tedious. Here's Lagrange multiplier method: For a function f(x, y), the extremum under a constraint g(x, y) = 0 is determined by defining L(x, y, λ) = f(x, y) - λg(x, y) and by evaluating OL/Ox = /y = /= 0. Using this method, determine the maximum value of f(x, y) = 2x + 3y under the constraint x² + y² 1 (hint: you need to check which extremum gives the maximum value).

(1) Lagrange multiplier is a very useful technique to determine the extremum of a function under constraint(s). For a function f(x, y), the extremum (minimum or maximum) is determined by evaluating of /ax = af/ay = 0. But determining the extremum under a constraint g(x, y) = 0 can be tricky. Even if y can be expressed explicitly in terms of x from g(x, y) = 0, the calculations may become tedious. Here's Lagrange multiplier method: For a function f(x, y), the extremum under a constraint g(x, y) = 0 is determined by defining L(x, y, λ) = f(x, y) - λg(x, y) and by evaluating OL/Ox = /y = /= 0. Using this method, determine the maximum value of f(x, y) = 2x + 3y under the constraint x² + y² 1 (hint: you need to check which extremum gives the maximum value).

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(1) Lagrange multiplier is a very useful technique to determine the extremum of a function under constraint(s). For a function f(x, y), the extremum (minimum or maximum) is determined by evaluating of /əx = əƒ/əy 0. But determining the extremum under a constraint g(x, y) = 0 can be tricky. Even if y can be expressed explicitly in terms of x from g(x, y) = 0, the calculations may become tedious. = Here's Lagrange multiplier method: For a function f(x, y), the extremum under a constraint g(x, y) = 0 is determined by defining L(x, y, λ) = f (x, y) − λg(x, y) and by evaluating OL/Əx = ƏL/Əy = ƏLƏλ = 0. Using this method, determine the maximum value of f(x,y) = 2x + 3y under the constraint x² + y² = 1 (hint: you need to check which extremum gives the maximum value).
Transcribed Image Text:(1) Lagrange multiplier is a very useful technique to determine the extremum of a function under constraint(s). For a function f(x, y), the extremum (minimum or maximum) is determined by evaluating of /əx = əƒ/əy 0. But determining the extremum under a constraint g(x, y) = 0 can be tricky. Even if y can be expressed explicitly in terms of x from g(x, y) = 0, the calculations may become tedious. = Here's Lagrange multiplier method: For a function f(x, y), the extremum under a constraint g(x, y) = 0 is determined by defining L(x, y, λ) = f (x, y) − λg(x, y) and by evaluating OL/Əx = ƏL/Əy = ƏLƏλ = 0. Using this method, determine the maximum value of f(x,y) = 2x + 3y under the constraint x² + y² = 1 (hint: you need to check which extremum gives the maximum value).
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