Problems

In Problems 1-10, find the maxima and minima.

In Problems 11-16, find the maximum and minimum subject to the given side conditions.

In Problems 17-26, determine whether the maxima and minima exist, and if so, find them.

27            Find three positive numbers x, y, and z such that x + y + z = 8 and x²yz is a maximum.

28            Find three positive numbers x, y, and z such that x + y + z = 100 and x²y²z is a maximum.

29            A package can be sent overseas by the air mail small packet rate if its length plus girth is at most 36 inches. Find the dimensions of the rectangular solid of maximum volume which can be sent by the small packet rate.

30            Find the volume of the largest rectangular solid which can be inscribed in a sphere of radius one.

31             Find the volume of the largest rectangular solid with faces parallel to the coordinate planes which can be inscribed in the ellipsoid x²/4 + y² + z²/9 = 1.

32            A triangle with sides a, b, c and perimeter p = a + b + c has area

Find the triangle of maximum area with perimeter p = 1.

33            Find the point on the plane x + 2y - z = 10 which is nearest to the origin.

34            Find the point on the plane x + y + z = 0 which is nearest to the point (l, 2, 3).

35            Find the points on the surface xyz = 1 which are nearest to the origin.

36            Find the point on the surface z = xy + 1 which is nearest to the origin.

37            Show that the rectangular solid with volume one and minimum surface area is the unit cube.

38            Show that the rectangular solid with surface area six and maximum volume is the unit cube.

39            A rectangular box with volume V in.³ is to be built with the sides and bottom made of material costing one cent per square inch, and the top costing two cents per square inch. Find the shape with the minimum cost.

40            A firm can produce and sell x units of one commodity and y units of another commodity for a profit of

P(x,y) = 100x + 200y - 10xy - x² - 500.

Due to limitations on plant capacity, x ≤ 10 and y ≤ 5. Find the values of x and y where the profit is a maximum.

41             x units of commodity one and y units of commodity two can be produced and sold at a profit of

P(x, y) = 400x + 500y - x² - y² - xy - 20000. Find the values of x and y where the profit is a maximum.

42            x units of commodity one can be produced at a cost of

C₁(x) = 1000 + 5x, and y units of commodity two can be produced at a cost of

C₂(y) = 2000 + 8y. Moreover, x units of one and y units of two can be sold for a total revenue of

Find the values of x and y where the profit is a maximum.

43            Suppose that with x man hours of labor and y units of capital, z = f(x, y) units of a commodity can be produced. The ratio z/x is called the average production per man hour. Show that ∂z/∂x = z/x when the average production per man hour is a maximum.

44            (Method of Least Squares) A straight line is to be fit as closely as possible to the set of three experimentally observed points (1,6), (2,9), and (3, 10). The line which best Jits these points is the line y = mx + b for which the sum of the squares of the errors,

E = [(m · 1 + b) - 6]² + [(m · 2 + h) - 9]² + [(m · 3 + b) - 10]²,

is a minimum. Find m and b such that E is a minimum.