Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.
Start Practicing

Worksheet: Finding the Stationary Points of Two Variable Functions

Q1:

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

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

Q2:

Find all stationary points of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ βˆ’ 3 π‘₯ + 𝑦 βˆ’ 3 𝑦 3 3 .

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

Q3:

Find all local maxima and minima of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ βˆ’ 3 π‘₯ + 𝑦 3 2 .

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

Q4:

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

  • A ο€Ό 1 , βˆ’ 1 2  is a local minimum point.
  • B ο€Ό βˆ’ 1 , 1 2  is a local maximum point.
  • C ο€Ό βˆ’ 1 , 1 2  is a saddle point.
  • D ο€Ό βˆ’ 1 , 1 2  is a local minimum point.
  • E ο€Ό 1 , βˆ’ 1 2  is a local maximum point.

Q5:

Find all stationary points of the function 𝑓 ( π‘₯ , 𝑦 ) = 2 π‘₯ βˆ’ 6 π‘₯ 𝑦 + 𝑦 3 2 , stating whether they are minima, maxima, or saddle points.

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

Q6:

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

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

Q7:

Find all stationary points of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ βˆ’ 1 2 π‘₯ + 𝑦 + 8 𝑦 3 2 , stating whether they are minima, maxima, or saddle points.

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

Q8:

Find all stationary points of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ + 3 π‘₯ + 𝑦 βˆ’ 3 𝑦 3 2 3 2 , stating whether they are minima, maxima, or saddle points.

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

Q9:

Find all stationary points of the function 𝑓 ( π‘₯ , 𝑦 ) = βˆ’ 4 π‘₯ + 4 π‘₯ 𝑦 βˆ’ 2 𝑦 + 1 6 π‘₯ βˆ’ 1 2 𝑦 2 2 , stating whether they are minima, maxima, or saddle points.

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