← Math 21a: Multivariable Calculus

Unit 4: Optimization in Several Variables

Finding hilltops and mountain passes with the gradient, then fencing the search onto a curve where Lagrange's tangency trick takes over.

9 topics · 29 questions

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4.1 Critical Points of a Function of Two Variables

  1. How it worksHow does setting both partial derivatives fx and fy equal to zero locate a function's critical points?
  2. Worked exampleFind the critical points of f(x, y) = x2 + y2 - 4x - 6y.
  3. Common mistakeWhy doesn't finding a critical point automatically mean the function has a maximum or a minimum there?
Your turn1 of 2
Find the x-coordinate of the critical point of f(x, y) = 4x2 + 4y2 - 10x - -5y.

4.2 The Second Derivative Test

  1. How it worksHow does the discriminant D = f_xx f_yy - (f_xy)2 combine curvature in different directions into a single test?
  2. Worked exampleShow how the second derivative test classifies the critical point of f(x, y) = x2 + y2 at (0, 0).
  3. Common mistakeWhy doesn't D = 0 automatically mean the critical point in question is a saddle point?
Your turn1 of 2
Find the discriminant D = f_xx·f_yy - (f_xy)2 at the origin for f(x, y) = 4x2 + 4y2 + 5xy.

4.3 Saddle Points

  1. How it worksHow does rotating a saddle-shaped surface like z = x2 - y2 reveal it curving upward along one axis and downward along the other, from the very same center point?
  2. Common mistakeWhy isn't a mountain pass, a real-world saddle point, either a mountain peak or a valley floor?
  3. Worked exampleShow that f(x, y) = x2 - y2 has a saddle point at the origin using the second derivative test.
Your turn1 of 2
Find the discriminant D = f_xx·f_yy - (f_xy)2 at the origin for f(x, y) = 4x2 - 4y2.

4.4 Global Extrema on Closed and Bounded Regions

  1. How it worksHow does checking both the interior critical points and the entire boundary curve locate a function's global extrema on a closed region?
  2. Worked exampleFind the maximum and minimum of f(x, y) = x2 + y2 on the closed disk x2 + y2 ≤ 4.
  3. Going furtherWhat happens to the search for global extrema when the region is unbounded instead of closed and bounded?
Your turn1 of 2
Find the maximum of f(x, y) = 4(x2+y2) + 7 on the closed disk x2 + y2 ≤ 19.

4.5 Optimization Word Problems

  1. How it worksHow do you turn a real-world constraint like a fixed volume into a single function of two variables ready for optimization?
  2. Common mistakeWhy isn't the first critical point you find automatically the answer to an optimization word problem?
  3. Worked exampleFind the dimensions of the open-top box with volume 32 cubic meters that has the minimum possible surface area.
Your turn1 of 2
An open-top box with a square base of side s has volume V = 73, so its height is V/s2. Write its surface area S(s) = s2 + 4·s·(height) as a function of s.

4.6 Constrained Optimization and Lagrange Multipliers

  1. How it worksHow do two families of level curves slide against each other until they touch tangentially, marking the constrained maximum?
  2. How it worksHow does the condition that the gradient of f runs parallel to the gradient of the constraint g turn into the Lagrange multiplier equations?
  3. Worked exampleFind the maximum of f(x, y) = xy subject to the constraint x + y = 10 using Lagrange multipliers.
  4. Common mistakeWhy doesn't the Lagrange multiplier lambda itself need any concrete physical meaning for the optimization method to work?
  5. Going furtherWhat happens to the tangency picture when the constraint curve just grazes a level curve instead of crossing through it?
Your turn1 of 2
Find the maximum of f(x, y) = 4·x·y subject to the constraint x + y = 7, using Lagrange multipliers.

4.7 Lagrange Multipliers with Two Constraints

  1. How it worksHow do two separate Lagrange multipliers combine the gradients of two constraints with the gradient of f?
  2. Worked exampleFind the extrema of f(x, y, z) = x + y + z subject to x2 + y2 = 1 and x + z = 1.
  3. Ties togetherHow does the two-constraint Lagrange condition generalize the single-constraint Lagrange multiplier method?
Your turn1 of 2
Find the maximum of f(x, y, z) = x + y + z subject to x2 + y2 = 14 and x + z = 3, using two Lagrange multipliers.

4.8 Least Squares and Data Fitting

  1. How it worksHow does turning the sum of squared errors into a function of slope and intercept let ordinary calculus optimize the best-fit line?
  2. Worked exampleFind the least-squares line through the points (0, 1), (1, 2), and (2, 4).
  3. Going furtherWhat happens to a least-squares fit when one data point sits far away from the general trend of the rest?
Your turn1 of 2
Find the slope of the least-squares line through the points (0, 5), (1, 4), and (2, 9).

4.9 Constrained Optimization Without Lagrange: the Substitution Method

  1. How it worksHow does substituting a constraint directly into the objective function turn a constrained problem into an ordinary single-variable optimization?
  2. Worked exampleFind the maximum of f(x, y) = xy subject to x + y = 10 by substituting y = 10 - x, and compare the answer to the Lagrange multiplier result.
  3. Going furtherWhat happens to the substitution method when the constraint equation cannot be solved cleanly for either variable?
Your turn1 of 2
Maximize f(x, y) = x·y subject to x + y = 14 by substituting y = 14 - x. Find the x-value at the maximum.