Optimization in two variables and Lagrange multipliers, question 11

Topic: Optimization in two variables and Lagrange multipliers
Difficulty: Standard
Suggested time: 4 min
Marks: 2
Theory: UBC Math 100 §16.3Key results

A student uses Lagrange multipliers to find the maximum value of the function below on the circle $x^{2} + y^{2} = 9$ , as shown. Find the first line that is wrong.

f(x, y) = x^{2} + 3y^{2}

$2x = 2\lambda x, \qquad 6y = 2\lambda y, \qquad x^{2} + y^{2} = 9$
The Lagrange conditions, with g(x, y) = x² + y².
$\lambda = 1$
Divide the first equation by 2x.
$6y = 2y \ \Rightarrow\ y = 0 \ \Rightarrow\ x = \pm 3$
Put λ = 1 into the second equation, then use the circle.
$f(\pm 3, 0) = 9 \quad\Longrightarrow\quad \text{maximum value } 9$
Every candidate gives 9, so the maximum is 9.

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Theory UBC Math 100 §16.3Key results

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Optimization in two variables and Lagrange multipliers practice question 11