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 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 + 2 𝑧 given that 4 𝑥 + 9 𝑦 3 6 𝑧 = 3 6 .

  • Amaximum: 8 9 , 0 , 4 , minimum: 8 9 , 0 , 4
  • Bmaximum: 9 8 , 5 9 4 , 1 4 , minimum: 9 5 , 0 , 2 5
  • Cmaximum: 9 8 , 5 9 4 , 1 4 , minimum: 9 5 , 0 , 2 5
  • Dmaximum: 8 9 , 0 , 4 , minimum: 8 9 , 0 , 4
  • Emaximum: 9 5 , 0 , 2 5 , minimum: 9 8 , 5 9 4 , 1 4

Q2:

Find all local maxima and minima of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 3 𝑥 + 𝑦 .

  • Alocal minimum at ( 1 , 0 ) , saddle point at ( 1 , 0 )
  • Blocal minimum at ( 0 , 1 ) , saddle point at ( 0 , 1 )
  • Clocal maximum at ( 1 , 0 ) , saddle point at ( 1 , 0 )
  • Dlocal minimum at ( 1 , 0 ) , saddle point at ( 1 , 0 )
  • Elocal maximum at ( 0 , 1 ) , saddle point at ( 0 , 1 )

Q3:

Find all stationary points of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 2 𝑦 , stating whether they are minima, maxima, or saddle points.

  • Alocal maximum at ( 1 , 2 )
  • BThe function has no stationary points.
  • Csaddle point at ( 1 , 2 )
  • Dlocal minimum at ( 1 , 2 )

Q4:

Find all stationary points of the function 𝑓 ( 𝑥 , 𝑦 ) = 4 𝑥 4 𝑥 𝑦 + 2 𝑦 + 1 0 𝑥 6 𝑦 , stating whether they are minima, maxima, or saddle points.

  • A 1 , 1 2 is a local minimum point.
  • B 1 , 1 2 is a local minimum point.
  • C 1 , 1 2 is a local maximum point.
  • D 1 , 1 2 is a saddle point.
  • E 1 , 1 2 is a local maximum point.

Q5:

Find all stationary points of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 3 𝑥 + 𝑦 3 𝑦 .

  • A ( 1 , 1 ) is a local minimum point, ( 1 , 1 ) and ( 1 , 1 ) are local maximum points, and ( 1 , 1 ) is a saddle point.
  • B ( 1 , 1 ) is a local minimum point, ( 1 , 1 ) is a local maximum point, and ( 1 , 1 ) and ( 1 , 1 ) are saddle points.
  • C ( 1 , 1 ) is a local minimum point, ( 1 , 1 ) is a local maximum point, and ( 1 , 1 ) and ( 1 , 1 ) are saddle points.
  • D ( 1 , 1 ) and ( 1 , 1 ) are local mimimum points, ( 1 , 1 ) is a local maximum point, and ( 1 , 1 ) is a saddle point.
  • E ( 1 , 1 ) is a local minimum point, ( 1 , 1 ) is a local maximum point, and ( 1 , 1 ) and ( 1 , 1 ) are saddle points.

Q6:

Find all stationary points of the function 𝑓 ( 𝑥 , 𝑦 ) = 4 𝑥 + 4 𝑥 𝑦 2 𝑦 + 1 6 𝑥 1 2 𝑦 , stating whether they are minima, maxima, or saddle points.

  • A ( 1 , 2 ) is a local minimum point.
  • B ( 1 , 2 ) is a saddle point.
  • C ( 1 , 2 ) is a local maximum point.
  • D ( 1 , 2 ) is a local minimum point.
  • E ( 1 , 2 ) is a local maximum point.

Q7:

Find all stationary points of the function 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 6 𝑥 𝑦 + 𝑦 , stating whether they are minima, maxima, or saddle points.

  • A ( 0 , 0 ) and ( 3 , 9 ) are saddle points.
  • B ( 0 , 0 ) is a local maximum point and ( 3 , 9 ) is a saddle point.
  • C ( 0 , 0 ) and ( 3 , 9 ) are local minimum points.
  • D ( 0 , 0 ) and ( 3 , 9 ) are local maximum points.
  • E ( 3 , 9 ) is a local minimum point and ( 0 , 0 ) is a saddle point.

Q8:

Find all stationary points of the function 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 + 6 𝑥 𝑦 + 3 𝑦 , stating whether they are minima, maxima, or saddle points.

  • A ( 0 , 0 ) is a local minimum point and ( 1 , 1 ) is a saddle point.
  • B ( 0 , 0 ) is a local maximum point and ( 1 , 1 ) is a saddle point.
  • C ( 1 , 1 ) is a local minimum point and ( 0 , 0 ) is a saddle point.
  • D ( 1 , 1 ) is a local maximum point and ( 0 , 0 ) is a saddle point.
  • E ( 1 , 1 ) is a local minimum point and ( 0 , 0 ) is a saddle point.

Q9:

Find all stationary points of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 1 2 𝑥 + 𝑦 + 8 𝑦 , stating whether they are minima, maxima, or saddle points.

  • A ( 4 , 2 ) is a local minimum point and ( 4 , 2 ) is a saddle point.
  • B ( 2 , 4 ) is a local minimum point and ( 2 , 4 ) is a saddle point.
  • C ( 4 , 2 ) is a local minimum point and ( 4 , 2 ) is a saddle point.
  • D ( 2 , 4 ) is a local minimum point and ( 2 , 4 ) is a saddle point.
  • E ( 2 , 4 ) is a local minimum point and ( 2 , 4 ) is a saddle point.

Q10:

Find all stationary points of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 3 𝑥 + 𝑦 3 𝑦 , stating whether they are minima, maxima, or saddle points.

  • A ( 0 , 2 ) and ( 2 , 2 ) are local minimum points, ( 2 , 0 ) is a local maximum point, and ( 0 , 0 ) is a saddle point.
  • B ( 0 , 2 ) is a local minimum point, ( 2 , 0 ) is a local maximum point, and ( 0 , 0 ) and ( 1 , 1 ) are saddle points.
  • C ( 0 , 2 ) is a local minimum point, ( 2 , 0 ) and ( 0 , 0 ) are local maximum points, and ( 2 , 2 ) is a saddle point.
  • D ( 2 , 0 ) is a local minimum point, ( 0 , 2 ) is a local maximum point, and ( 0 , 0 ) and ( 2 , 2 ) are saddle points.
  • E ( 0 , 2 ) is a local minimum point, ( 2 , 0 ) is a local maximum point, and ( 0 , 0 ) and ( 2 , 2 ) are saddle points.

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