Exercises:

Find the local extrema and saddle points of the following functions:
1.
f( x,y) = x2+4y2
2.
f( x,y) = x2-3y2
3.
f( x,y) = x2+xy+3x+2y
4.
f( x,y) = y2+xy-2x-2y
5.
f( x,y) = x2-4xy+y2+6y
6.
f( x,y) = -x2+2xy-y2+3x+4
7.
f( x,y) = 3x2+6xy+7y2-2x+4y
8.
f(x,y) = 4x2-6xy+5y2-20x+26y
9.
f( x,y) = x3-3x2+y2
10.
f( x,y) = x4+y4-y2
11.
f( x,y) = x3+3xy+y3
12.
f( x,y) = x3+6xy+y3
13.
f( x,y) = 4xy-x4-y4
14.
f( x,y) = x4+y4+4xy
15.
f( x,y) = x4+2x2y-2y
16.
f( x,y) = x4-2x2y+2y
17.
f( x,y) = sin( x) +cos( y)
18.
f( x,y) = xln( x) +yln( y)
19.
f( x,y) = xsin( y)
20.
f(x,y) = e2xcos( y)

Find the slope and y-intercept of the least squares line for each of the following data sets:
21.
( 1,1) , ( 2,2) , ( 3,3), (4,4)
   
22.
( 1,72) , ( 2,77) , (3,83), (4,97)
23.
( -1,1.2) , ( -2,2.3) , (-3,3.4)
24.
( 1,1) , ( 2,1) , ( 3,1)
25.
( 1,75) , ( 2,79) ,  ( 3,85),  ( 4,81)
26.
( 1,75) , (2,79) ,  ( 3,81) ,  ( 4,85)

27. Find the point in the plane z = x - y +1 which is closest to the origin. (Hint: minimize the square of the distance from a point ( x,y,x-y+1) to the origin ( 0,0,0) ).

28. Find the point in the plane z = x+2y+3 which is closest to the origin.

29. Find the point in the plane z = x+y closest to the point ( 2,2,1).

30. Find the point(s) on the surface z = ( x-1)2+y2 closest to the origin.

31. What dimensions of a rectangular box with a surface area of 64 in2 lead to a maximum volume?

32. What dimensions of a rectangular box with a volume of 64 in3 lead to a minimum surface area?

           

33. Acme sporting goods collected the following set of data relating price charged for a racket, x, to the number of rackets per week sold at that price. 

x = price  $50 $55 $60 $65
y = weekly sales 18 15 10 6

Fit this data to a linear demand function y = mx+b.

34. Repeat exercise 33 given the data set  

x = price  $50 $55 $60 $65
y = weekly sales 24 21 18 15

35.  What is the slope and y-intercept of the least squares line through the points ( x1,y1) , ( x2,y2), ( x3,y3) , and ( x4,y4) ?

36.  Show that the least squares line passes through the point (x*,y*), where x* is the average of the x coordinates of the data and y* is the average of the y-coordinates of the data.

37.  A house with width x, length y, and height z is to have a roofline with a height of 25¢.

If the house is to have a total floor space of 2000 ft2, what values of x,y, and z minimize the sum of the area of the sides and the roof.

38. If the roof costs 3 times more than the sides of the house, then what values of x,y,z in problem 29 minimize the cost of the house?

39. Find the point(s) on the surface z = 1-x2-y2 closest to the origin. Why are there infinitely many of them?  

40. A "ray" travels in a line from the point ( 4,2) to the x-axis, is reflected in a straight line to the y-axis, and then is reflected again to the point ( 2,3) .

What is the shortest possible path for the ray to travel in this manner from ( 4,2) to ( 2,3) ?

41. Write to Learn: Write a short essay which shows that if f( x,y) has a critical point at ( p,q) , then f( x,y) has a local minimum at ( p,q) only if fxx(p,q) > 0 and
D = fxx( p,q) fyy( p,q) -fxy2(p,q) > 0

42. Discussion: Explain why every point on the unit circle is a critical point of
f( x,y) = ( x2+y2-1) 2
Does f(x,y) have all saddle points on the unit circle, or does it have all minima on the unit circle? Explain.

43.  *Suppose a house with a height of 10 feet at each corner is to have a total floor space of 2000 ft2, and suppose that ps is the cost per square foot of the sides and that pr is the cost per square foot of the roof.

What width x, height y, and pitch of the roof q minimize the cost of the house? What shape should the house have if ps £ pr? What should the dimensions of the house be if ps = 2pr?