# This week: 12.4–6 webAssign: 12.4–6, due 2/15 11:55 p.m. Friday: ```MATH 251 – LECTURE 11
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 12.4–6
webAssign: 12.4–6, due 2/15 11:55 p.m.
Friday:
Kevin.
Next week: 12.7
webAssign: 12.7, opens 2/15 12 a.m.
Friday 2/19:
Midterm 1: Covering chapters 11 and 12.
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Directional derivatives
The surface z = f (x, y) can be viewed as a level curve of the function F (x, y, z) = z − f (x, y).
Exercise 1. If you are at the point (x0, y0, f (x0, y0)), in which direction should you move so that the function
F increases the fastest?
Exercise 2. Recall from previous lectures: what is the normal vector of the surface z = f (x, y) at the point
(x0, y0, f (x0, y0))?
The gradient of F (x, y, z) is always a normal vector to the level curves of the function F .
Exercise 3. Let f (x, y) = x2 + y 2. Sketch the level curve of f passing through the point (1, 1). What is
∇f (1, 1)?
Local maximum and minimum
Definition 4. 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 5. 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 6. 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 7. 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).
Exercise 8. Find all critical points of the function f (x, y) = x2 + xy + 4y.
Local maximum and minimum
Exercise 9. Find all critical points of the function f (x, y) = ex
3 +4xy
.
Local maximum and minimum
Assuming that f is smooth at the critical point (a, b), then there are three types of critical points.
Local maxima
Local minima
Local maximum and minimum
Theorem 10 (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) &gt; 0 and fxx
(a, b) &lt; 0 then (a, b) is a local maximum.
00
2) if D(a, b) &gt; 0 and fyy
(a, b) &gt; 0 then (a, b) is a local minimum.
3) if D(a, b) &lt; 0 then (a, b) is a saddle point.
Exercise 11. Find and classify the critical points of f (x, y) = x2 + y 2.
Local maximum and minimum
Exercise 12. Find and classify the critical points of f (x, y) = −x2 − y 2.
Exercise 13. Find and classify the critical points of f (x, y) = x2 − y 2.
Local maximum and minimum
Exercise 14. Find the point on the plane x − y + z = 7 that is closest to (1, 4, 6).
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