1. A contour diagram of a function f(x,y) is given below. The function has critical points at (0,0) and (2,2). Classify each of them as a local minimum, maximum or a saddle point. Explain your answer.
2. Find the global maximum of the function f(x,y) = y2 + x2 in the region R: 0<=x<=1, 0<=y<=2. Explain your reasoning.
3. Find local minima, maxima
and saddle points for the function f(x,y)
= 2y3 + 3 x2
- 6xy.
4. Let f(x,y) = 3xy - x2 - y3 .Find all critical points, local minima, maxima and saddle points of f(x,y).
5. A cruise missile has a remote
guidance device which is sensitive to both temperature and humidity. Army
engineers have worked out a formula to show the range at which the missile
can be controlled, f(t,h),
in
miles, as a function of the temperature
t,
in degrees Fahrenheit, and
percent humidity h:
f(t,h) = 10,000 - t2 - 2 ht - 2 h2
+ 200 t + 260h.
What are the optimal atmospheric conditions for controlling the missile ?
6. Let a function f(x,y)
and
a point (a,b)
be
such that
f x (a,b) = f
y(a,b)
= 0, f
xx (a,b) = 0, f
yy(a,b)
> 0, f
xy (a,b) < 0.
(a) What
can you conclude about the behavior of f(x,y)
near
(a,b)
?
(b) Sketch
a possible contour diagram for f(x,y)
near
(a,b).
7. Consider the integral
(a) Sketch
and label the region over which the integration is being performed.
(b) Rewrite
the integral with the integration performed in the opposite order.
8. Convert the integral
