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+2𝑧 given that 4𝑥+9𝑦36𝑧=36.

  • Amaximum: 89,0,4, minimum: 89,0,4
  • Bmaximum: 98,594,14, minimum: 95,0,25
  • Cmaximum: 95,0,25, minimum: 98,594,14
  • Dmaximum: 89,0,4, minimum: 89,0,4
  • Emaximum: 98,594,14, minimum: 95,0,25

Q2:

Find all local maxima and minima of the function 𝑓(𝑥,𝑦)=𝑥3𝑥+𝑦.

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

Q3:

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

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

Q4:

Find all stationary points of the function 𝑓(𝑥,𝑦)=4𝑥4𝑥𝑦+2𝑦+10𝑥6𝑦, stating whether they are minima, maxima, or saddle points.

  • A1,12 is a saddle point.
  • B1,12 is a local minimum point.
  • C1,12 is a local minimum point.
  • D1,12 is a local maximum point.
  • E1,12 is a local maximum point.

Q5:

Find all stationary points of the function 𝑓(𝑥,𝑦)=𝑥3𝑥+𝑦3𝑦.

  • A(1,1) is a local minimum point, (1,1) is a local maximum point, and (1,1) and (1,1) are saddle points.
  • B(1,1) and (1,1) are local mimimum points, (1,1) is a local maximum point, and (1,1) is a saddle point.
  • 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) is a local minimum point, (1,1) and (1,1) are local maximum points, 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)=4𝑥+4𝑥𝑦2𝑦+16𝑥12𝑦, stating whether they are minima, maxima, or saddle points.

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

Q7:

Find all stationary points of the function 𝑓(𝑥,𝑦)=2𝑥6𝑥𝑦+𝑦, stating whether they are minima, maxima, or saddle points.

  • A(0,0) and (3,9) are local minimum points.
  • B(0,0) is a local maximum point and (3,9) is a saddle point.
  • C(0,0) and (3,9) are saddle 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)=2𝑥+6𝑥𝑦+3𝑦, stating whether they are minima, maxima, or saddle points.

  • A(1,1) is a local minimum point and (0,0) is a saddle point.
  • B(0,0) is a local minimum point and (1,1) is a saddle point.
  • C(1,1) is a local maximum point and (0,0) is a saddle point.
  • D(0,0) is a local maximum point and (1,1) 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)=𝑥12𝑥+𝑦+8𝑦, stating whether they are minima, maxima, or saddle points.

  • A(2,4) is a local minimum point and (2,4) is a saddle point.
  • B(4,2) is a local minimum point and (4,2) is a saddle point.
  • C(2,4) is a local minimum point and (2,4) is a saddle point.
  • D(4,2) is a local minimum point and (4,2) 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+3𝑥+𝑦3𝑦, 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(2,0) is a local minimum point, (0,2) is a local maximum point, and (0,0) and (2,2) are saddle points.
  • D(0,2) is a local minimum point, (2,0) is a local maximum point, and (0,0) and (2,2) are saddle points.
  • E(0,2) is a local minimum point, (2,0) and (0,0) are local maximum points, and (2,2) is a saddle point.

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