Vector Calculus Independent Study

Unit 4: Maximization and Minimization

To find the maximium and minimum points of a function f(x⃗) on a set D, follow this procedure:

1. Look at all the critical points x⃗ where ∇f(x⃗) = 0. Throw out any that aren’t in D.

2. Look at any points where f is non-differentiable.

3. Look at the boundary of D. You can do this either by

• Parameterizing the boundary so that you have an unconstrained max/min problem, or by

• Lagrange multipliers: if the boundary of D is described as a level set by g(x⃗) = 0, then the critical points of f constrained to be on the boundary of D can be found by solving ∇f = λ∇g.

4. If ∇h(x⃗) = 0, then you can tell if x⃗ is a local max, min, or saddle point by looking at the Hessian:

H(f) = [ \begin{matrix}\frac{∂^{2}f}{∂x_{1}^{2}} & \frac{∂^{2}f}{∂x_{1}∂x_{2}} & … & \frac{∂^{2}f}{∂x_{1}∂x_{n}} \\ ⋮ & ⋮ & ... & ⋮ \\ \frac{∂^{2}f}{∂x_{n}∂x_{1}} & \frac{∂^{2}f}{∂x_{n}∂x_{2}} & … & \frac{∂^{2}f}{∂x_{n}^{2}} \end{matrix} ]

If all of the upper-left square sub-matrices of the Hessian have positive determinant, then the critical point is a local minimum. If the determinants alternate in sign starting with \frac{∂^{2}f}{∂x_{1}^{2}} < 0, then the critical point is a local maximum. If any of the determinants are zero, then the test is indeterminant. Finally, if none of the above ifs apply, you have a saddle point.

Suggested Procedure:

1. Read and do some problems from • Rogers Chapters 12 and 13,

• Marsden and Tromba 3rd edition chapter 4,

• Marsden and Tromba 4th edition chapter 3, or • Simmons, sections 19.7 and 19.8.

2. Take the sample test.

3. Take a unit test.