Harvard University, Math 20 Fall 2011, Instructor: Rachel Epstein
1
Review for Final Exam
You should review all the material for the midterms, as well as the following
new material below.
1. Optimization without constraints (Chapter 17.49):
(a) Know what a stationary point is and that these are the only possible
interior points that are (local or global) extreme points.
(b) Know and use the secondderivative test (Theorem 17.5), which helps
to determine if a stationary point is a local max/min or saddle point.
(c) Know what it means geometrically for a
set
to be convex.
(d) Know what it means geometrically for a
function
to be convex or
concave. (Ignore Jensen’s Inequality)
(e) Theorem 17.6 on page 628 is useful, but not necessary to know for
the test. You do not need to know Theorem 17.7, either.
(f) Know Theorem 17.8.
It is particularly useful in combination with
the theorems about determining concavity/convexity in sections 17.8
and 17.9.
(g) Know Theorem 17.9. Notice 17.11 follows from Theorems 17.9 and
17.8. You should know 17.11, but it should be obvious if you know
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 Fall '11
 RachelEpstein
 Math, Critical Point, Optimization, saddle point, Stationary point, Lagrange multiplier method

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