MATH 251 – LECTURE 12
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 12.7
webAssign: 12.7, due 2/22 11:55 p.m.
Friday 2/19:
Midterm 1: Covering chapters 11 and 12.
Next week: 13.1–4
webAssign: 13.1–3, opens 2/22 12 a.m.
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–4 p.m.
or by appointment.
Local maximum and minimum
Definition 1. A function f (x, y) is said to have a local maxima at a point (a, b) if f (a, b) ≥ f (x, y) for all
(x, y) close to (a, b).
Definition 2. A function f (x, y) is said to have a local minima at a point (a, b) if f (a, b) ≤ f (x, y) for all
(x, y) close to (a, b).
Theorem 3. If f (x, y) is differentiable at a local maximum (or minimum) (a, b), then ∇f (a, b) = h0, 0i.
Local maximum and minimum
Definition 4. A point (a, b) is said to be a critical point of f (x, y) if either ∇f (a, b) = h0, 0i, or one of the
first partial derivatives of f does not exists at (a, b).
Assuming that f is smooth at the critical point (a, b), then there are three types of critical points.
Local maxima
Local minima
Saddle points
Local maximum and minimum
Theorem 5 (Second derivative test). Assume that f is twice differentiable at a critical point (a, b). Let
00
00
fxx(a, b) fxy
(a, b) 00
00
00
D(a, b) = 00
= fxx
(a, b)fyy
(a, b) − (fxy
(a, b))2
00
fyx(a, b) fyy (a, b)
00
1) if D(a, b) > 0 and fxx
(a, b) < 0 then (a, b) is a local maximum.
00
2) if D(a, b) > 0 and fyy
(a, b) > 0 then (a, b) is a local minimum.
3) if D(a, b) < 0 then (a, b) is a saddle point.
Exercise 6. Find and classify the critical points of f (x, y) = x2 − y 2.
Maximum and minimum
Definition 7. A set S in R2 is said to be compact if it is closed and bounded.
Maximum and minimum
Typically, a compact set is the boundary and the interior of a smooth curve defined in implicit form g(x, y) = 0.
Maximum and minimum
Theorem 8. If S is compact and f is continuous on S, then f obtains its extremal values on S.
Example 9. Consider f (x) = x on [0, 1] and on (0, 1).
Maximum and minimum
How do we find extremal values?
1) Find critical points of f in the interior of the set S.
2) Find the extremal values of f on the boundary of S.
3) Compare!
Lagranges method
Recall that a level curve (or surface) of a function f (x, y, z) is a curve (or surface) defined by f (x, y, z) = k for
some constant k.
Exercise 10. Draw the level curves of the function f (x, y) = x2 + y 2.
Lagranges method
Exercise 11. Find the point on the line x + 6y = 2 which is closes to the origin.
Lagranges method
Exercise 11. Find the point on the line x + 6y = 2 which is closes to the origin.
Lagranges method
Exercise 12. Minimize the function f (x, y) = x2 + 2y 2 subject to the constraints that x + y 2 + 41 = 0.
Lagranges method
Exercise 12. Minimize the function f (x, y) = x2 + 2y 2 subject to the constraints that x + y 2 + 41 = 0.
Lagranges method
Exercise 13. Find the point on the plane x − y + z = 7 that is closest to (1, 4, 6).
Lagranges method
Exercise 13. Find the point on the plane x − y + z = 7 that is closest to (1, 4, 6).