
# 13.9E: Optimization of Functions of Several Variables (Exercises)


## Absolute Extrema on Closed and Bounded Regions

In exercises 1 - 4, find the absolute extrema of the given function on the indicated closed and bounded set $$R$$.

1) $$f(x,y)=xy−x−3y; R$$ is the triangular region with vertices $$(0,0),(0,4),$$ and $$(5,0)$$.

2) Find the absolute maximum and minimum values of $$f(x,y)=x^2+y^2−2y+1$$ on the region $$R=\{(x,y)∣x^2+y^2≤4\}.$$

$$(0,1,0)$$ is the absolute minimum and $$(0,−2,9)$$ is the absolute maximum.

3) $$f(x,y)=x^3−3xy−y^3$$ on $$R=\{(x,y):−2≤x≤2,−2≤y≤2\}$$

4) $$f(x,y)=\frac{−2y}{x^2+y^2+1}$$ on $$R=\{(x,y):x^2+y^2≤4\}$$

There is an absolute minimum at $$(0,1,−1)$$ and an absolute maximum at $$(0,−1,1)$$.

## Applications

5) Find three positive numbers the sum of which is $$27$$, such that the sum of their squares is as small as possible.

6) Find the points on the surface $$x^2−yz=5$$ that are closest to the origin.

Hint:
Use the distance formula. But note that you can leave off the square root, since the minimum value of the square of the distance will also minimize the distance.
$$(\sqrt{5},0,0),(−\sqrt{5},0,0)$$

7) Find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on the line $$x+y+z=1$$.

8) The sum of the length and the girth (perimeter of a cross-section) of a package carried by a delivery service cannot exceed $$108$$ in. Find the dimensions of the rectangular package of largest volume that can be sent.

$$18$$ by $$36$$ by $$18$$ in.

9) A cardboard box without a lid is to be made with a volume of $$4$$ ft3. Find the dimensions of the box that requires the least amount of cardboard.

10) Find the point on the surface $$f(x,y)=x^2+y^2+10$$ nearest the plane $$x+2y−z=0.$$ Identify the point on the plane.

Hint:
Here one approach is the find the distance between a point $$(x_0, y_0, z_0)$$ on the surface and the plane, using what you learned in Section 11.5. Then you can substitute the surface function into this distance function for $$z_0$$ and substitute $$x$$ for $$x_0$$ and $$y$$ for $$y_0$$. This will give you a function of $$x$$ and $$y$$ that you can minimize.
$$(0.5,1,11.25)$$ is the point on the surface nearest the plane.
Although it was not requested, note that $$(\frac{47}{24},\frac{47}{12},\frac{235}{24})$$ is the point on the plane that is nearest the surface.
See this problem illustrated in CalcPlot3D.

11) Find the point in the plane $$2x−y+2z=16$$ that is closest to the origin.

12) A company manufactures two types of athletic shoes: jogging shoes and cross-trainers. The total revenue from $$x$$ units of jogging shoes and $$y$$ units of cross-trainers is given by $$R(x,y)=−5x^2−8y^2−2xy+42x+102y,$$ where $$x$$ and $$y$$ are in thousands of units. Find the values of $$x$$ and $$y$$ to maximize the total revenue.

$$x=3$$ and $$y=6$$

13) A shipping company handles rectangular boxes provided the sum of the length, width, and height of the box does not exceed $$96$$in. Find the dimensions of the box that meets this condition and has the largest volume.

14) Find the maximum volume of a cylindrical soda can such that the sum of its height and circumference is $$120$$ cm.

$$V=\frac{64,000}{π}\,\text{cm}^3≈20,372\,\text{cm}^3$$