Lesson test: Lagrange multipliers

Lagrange multipliers

1.[2p]
Use Lagrange multipliers to find the maximum of $f (x, y) = x y$ subject to $x + 2 y = 20$ .

CorrectNot quite: 50

2.[2p]
Find the maximum of $f (x, y) = 3 x + 4 y$ on the circle $x^{2} + y^{2} = 1$ .

CorrectNot quite: 5

3.[2p]
Find the minimum of $x^{2} + y^{2} + z^{2}$ on the plane $x + 2 y + 2 z = 12$ .

CorrectNot quite: 16

4.[3p]
An open-topped box must hold $32$ cubic metres. What is the least total area, in square metres, of its base and four sides?

CorrectNot quite: 48

5.[1p]
At a point where $f$ has a maximum on the constraint curve $g (x, y) = c$ , and $\nabla g \neq 𝟎$ there, how does the level curve of $f$ through the point meet the constraint curve?

It is tangent to the constraint curve

Correct

It crosses the constraint curve at right angles

The answer is: It is tangent to the constraint curve

It crosses the constraint curve at an angle that depends on $λ$

The answer is: It is tangent to the constraint curve

It does not meet the constraint curve, because $\nabla f = 𝟎$ there

The answer is: It is tangent to the constraint curve

6.[3p]
A firm produces $Q = 2 \sqrt{L K}$ from labour $L$ at £1 a unit and capital $K$ at £4 a unit, with a budget of £400. What is the Lagrange multiplier $λ$ at the optimum, the extra output from one more pound of budget?

CorrectNot quite: 0.5

7.[2p]
The maximum of $x y$ subject to $x + y = 10$ is $25$ , with multiplier $λ = 5$ . Use $λ$ to estimate the maximum of $x y$ subject to $x + y = 10.2$ . Give it to two decimal places.

CorrectNot quite: 26

8.[3p]
The plane $x + y + z = 4$ cuts the cylinder $x^{2} + y^{2} = 8$ in an ellipse. What is the largest value of $z$ on the ellipse?

CorrectNot quite: 8

9.[3p]
Which of these statements about the method of Lagrange multipliers are true?

Select all that apply

A solution of $\nabla f = λ \nabla g$ , $g = c$ need not be a maximum or a minimum of $f$ on the constraint

Correct

A constrained extremum can occur at a point where $\nabla g = 𝟎$ and be missed by the equations

Correct

The multiplier $λ$ can be negative

Correct

At a constrained maximum, $\nabla f$ must be $𝟎$

The answer is: A solution of $\nabla f = \lambda \nabla g$, $g = c$ need not be a maximum or a minimum of $f$ on the constraint, A constrained extremum can occur at a point where $\nabla g = \mathbf{0}$ and be missed by the equations, The multiplier $\lambda$ can be negative

The multiplier $λ$ is always $1$ when the constraint is linear

1.[2p] Use Lagrange multipliers to find the maximum of f(x,y)=xy subject to x+2y=20.

2.[2p] Find the maximum of f(x,y)=3x+4y on the circle x2+y2=1.

3.[2p] Find the minimum of x2+y2+z2 on the plane x+2y+2z=12.

4.[3p] An open-topped box must hold 32 cubic metres. What is the least total area, in square metres, of its base and four sides?

5.[1p] At a point where f has a maximum on the constraint curve g(x,y)=c, and ∇g≠𝟎 there, how does the level curve of f through the point meet the constraint curve?

6.[3p] A firm produces Q=2LK from labour L at £1 a unit and capital K at £4 a unit, with a budget of £400. What is the Lagrange multiplier λ at the optimum, the extra output from one more pound of budget?

7.[2p] The maximum of xy subject to x+y=10 is 25, with multiplier λ=5. Use λ to estimate the maximum of xy subject to x+y=10.2. Give it to two decimal places.

8.[3p] The plane x+y+z=4 cuts the cylinder x2+y2=8 in an ellipse. What is the largest value of z on the ellipse?

9.[3p] Which of these statements about the method of Lagrange multipliers are true?

1.[2p]
Use Lagrange multipliers to find the maximum of $f (x, y) = x y$ subject to $x + 2 y = 20$ .

2.[2p]
Find the maximum of $f (x, y) = 3 x + 4 y$ on the circle $x^{2} + y^{2} = 1$ .

3.[2p]
Find the minimum of $x^{2} + y^{2} + z^{2}$ on the plane $x + 2 y + 2 z = 12$ .

4.[3p]
An open-topped box must hold $32$ cubic metres. What is the least total area, in square metres, of its base and four sides?

5.[1p]
At a point where $f$ has a maximum on the constraint curve $g (x, y) = c$ , and $\nabla g \neq 𝟎$ there, how does the level curve of $f$ through the point meet the constraint curve?

6.[3p]
A firm produces $Q = 2 \sqrt{L K}$ from labour $L$ at £1 a unit and capital $K$ at £4 a unit, with a budget of £400. What is the Lagrange multiplier $λ$ at the optimum, the extra output from one more pound of budget?

7.[2p]
The maximum of $x y$ subject to $x + y = 10$ is $25$ , with multiplier $λ = 5$ . Use $λ$ to estimate the maximum of $x y$ subject to $x + y = 10.2$ . Give it to two decimal places.

8.[3p]
The plane $x + y + z = 4$ cuts the cylinder $x^{2} + y^{2} = 8$ in an ellipse. What is the largest value of $z$ on the ellipse?

9.[3p]
Which of these statements about the method of Lagrange multipliers are true?