# Worksheet: Extrema of a Multivariable Function

In this worksheet, we will practice finding critical points of a multivariable function and identifying the local maximum, local minimum, and saddle points of the function.

Q1:

Find the constrained maxima and minima of given that .

• Amaximum: , minimum:
• Bmaximum: , minimum:
• Cmaximum: , minimum:
• Dmaximum: , minimum:
• Emaximum: , minimum:

Q2:

Find all local maxima and minima of the function .

• Alocal minimum at , saddle point at
• Blocal maximum at , saddle point at
• Clocal minimum at , saddle point at
• Dlocal minimum at , saddle point at
• Elocal maximum at , saddle point at

Q3:

Find all stationary points of the function , stating whether they are minima, maxima, or saddle points.

• Alocal minimum at
• CThe function has no stationary points.
• Dlocal maximum at

Q4:

Find all stationary points of the function , stating whether they are minima, maxima, or saddle points.

• A is a saddle point.
• B is a local minimum point.
• C is a local minimum point.
• D is a local maximum point.
• E is a local maximum point.

Q5:

Find all stationary points of the function .

• A is a local minimum point, is a local maximum point, and and are saddle points.
• B and are local mimimum points, is a local maximum point, and is a saddle point.
• C is a local minimum point, is a local maximum point, and and are saddle points.
• D is a local minimum point, and are local maximum points, and is a saddle point.
• E is a local minimum point, is a local maximum point, and and are saddle points.

Q6:

Find all stationary points of the function , stating whether they are minima, maxima, or saddle points.

• A is a saddle point.
• B is a local minimum point.
• C is a local minimum point.
• D is a local maximum point.
• E is a local maximum point.

Q7:

Find all stationary points of the function , stating whether they are minima, maxima, or saddle points.

• A and are local minimum points.
• B is a local maximum point and is a saddle point.
• C and are saddle points.
• D and are local maximum points.
• E is a local minimum point and is a saddle point.

Q8:

Find all stationary points of the function , stating whether they are minima, maxima, or saddle points.

• A is a local minimum point and is a saddle point.
• B is a local minimum point and is a saddle point.
• C is a local maximum point and is a saddle point.
• D is a local maximum point and is a saddle point.
• E is a local minimum point and is a saddle point.

Q9:

Find all stationary points of the function , stating whether they are minima, maxima, or saddle points.

• A is a local minimum point and is a saddle point.
• B is a local minimum point and is a saddle point.
• C is a local minimum point and is a saddle point.
• D is a local minimum point and is a saddle point.
• E is a local minimum point and is a saddle point.

Q10:

Find all stationary points of the function , stating whether they are minima, maxima, or saddle points.

• A and are local minimum points, is a local maximum point, and is a saddle point.
• B is a local minimum point, is a local maximum point, and and are saddle points.
• C is a local minimum point, is a local maximum point, and and are saddle points.
• D is a local minimum point, is a local maximum point, and and are saddle points.
• E is a local minimum point, and are local maximum points, and is a saddle point.