(i) Find the critical points of f (x) in [a, b].(ii) Evaluate f (x) at the critical points and the endpoints a and b.(iii) The largest and smallest values are the extreme values of f (x) on [a, b].

Optimization plays a role in a wide range of disciplines, including the physical sciences, economics, and biology. For example, scientists have studied how migrating birds choose an optimal velocity v that maximizes the distance they can travel without stopping, given the energy that can be stored as body fat.

In many optimization problems, the first step is to write down the objective function. This is the function whose minimum or maximum we need. Once we find the objective function, we can apply the techniques developed in this chapter. Our first examples require optimization on a closed interval [a, b]. Let’s recall the steps for finding extrema developed in Section 4.2:

Read Only

Rectangle

2

Secondary Function

Secondary Function

Objective Function:

Objective

2 2

Use the to write the in terms of one variFun able.

12

c i12

t on

P L x y

A x x L x Lx x

A piece of wire of length L is bent into the shape of a rectangle. Which dimensions produce the rectangle of maximum area?

y

We need to

optimize a

Real Domai

rea o 0

n

.2

2

:

0

,n

L

L

x

A x

Conjecture???

A piece of wire of length L is bent into the shape of a rectangle. Which dimensions produce the rectangle of maximum area?

21Maximize on 0,2 2

LA x Lx x

y

is a critical poin1'4

2 t02

A x LxL x

gives the maximum area4

'' 24

LALA

max area occurs with a square, such that .4Lx

2nd Derivative Test

x

Minimizing Travel Time Your task is to build a road joining a ranch to a highway that enables drivers to reach the city in the shortest time. How should this be done if the speed limit is 60 km/h on the road and 110 km/h on the highway? The perpendicular distance from the ranch to the highway is 30 km, and the city is 50 km down the highwayThis problem is more complicated than the previous one, so we’ll analyze it in three steps. You can follow these steps to solve other optimization problems.

Step 1. Choose variables.

Step 2. Find the objective function and the interval.

? dT x d vt tv

2 230 50 , in 0,5060 110x xT x x

Step 3. Optimize.

x

City

Ranch

QP

302 230 x

x 50 x

50

Add the constantsinto our diagram

1/ 2 22

2 2

2 2 2

1 2 1 1' 0110 11060 900120 900

110 60 900 11 6 900

121 36 900 85 32,400

19.5237

x xT xxx

x x x x

x x x

x

2 230 50 , in 0,5060 110x xT x x

1/ 221 5 190060 11 110

T x x x

Minimizing Travel Time Your task is to build a road joining a ranch to a highway that enables drivers to reach the city in the shortest time. How should this be done if the speed limit is 60 km/h on the road and 110 km/h on the highway? The perpendicular distance from the ranch to the highway is 30 km, and the city is 50 km down the highway.

T xx019.523750

0.955 h0.874 h0.972 h

19.524 kmx

& cross-multiply

Minimizing Travel Time Your task is to build a road joining a ranch to a highway that enables drivers to reach the city in the shortest time. How should this be done if the speed limit is 60 km/h on the road and 110 km/h on the highway? The perpendicular distance from the ranch to the highway is 30 km, and the city is 50 km down the highway

19.524 kmx

2 2

1 130 25 120 55 12040 40

1 155 6600 3 58 660040 40

P r r r r r

P r r r r r r

Optimal Price All units in a 30-unit apartment building are rented out when the monthly rent is set at r = $1000/month. A survey reveals that one unit becomes vacant with each $40 increase in rent. Suppose that each occupied unit costs $120/month in maintenance. Which rent r maximizes monthly profit?

Do monthlyprofit has b

all unieen opt

tsim

riz

enteded???

Profit # of rented units Income generated per rented unitStep 1. Choose variables.

Monthly profit in terms of the cost to rent a unitP r

100030 12040

rP r r

Step 2. Find the objective function and the interval.

in [1000,2200)r

If we raise the rent, all of our units may not be rented; there is a cost per rental unit

What rent brings our numberof rented units to zero?

1 5820r

Optimal Price All units in a 30-unit apartment building are rented out when the monthly rent is set at r = $1000/month. A survey reveals that one unit becomes vacant with each $40 increase in rent. Suppose that each occupied unit costs $120/month in maintenance. Which rent r maximizes monthly profit?

21 58 660040

P r r r 1000 2200r

1' 58 020

P r r P rr

10001160

$26, 400$27,040

the rent will be $1160

Step 3. Optimize.

1160r

1"20

P r

Design a cylindrical can of volume 900 cm3 so that it uses the least amount of metal.

22

2 190 2 12 02 800A r r rr rr

, , 900 secondaryA r h V

22

900, 900V r h r h hr

0lim , lim

must be a min.r rA r A r

A c

Step 2. Find the objective function and the interval.

2, 2 2A r h r rh

We'll minimize 00 on ,A rr Step 3. Optimize.

32 2

1/3

1800 1800 1800 450' 4 0 4

0

445 cm

A r r r rr r

r

10.464 cmh

Step 1. Choose variables.

3900 cmV

Let's use limits to conclude will be a min.A c

Can we maximize ?A

The Principle of Least Distance states that a light beam reflected in a mirror travels along the shortest path. More precisely, a beam traveling from A to B, is reflected at the point P for which the path APB has minimum length. In the next example, we show that this minimum occurs when the angle of incidence is equal to the angle of reflection, that is, θ1 = θ2.

1/21/2 22 2 21 2

2 2 2 21 2

2 2 2 21 2

1 1 22

1 1' 2 2 12 2

0

cos cos

f x x h x L x h L x

x L x

x h L x h

x L x

x h L x h

Show that if P is the point for which the path APB has minimal length, then θ1 = θ2.

22 2 21 2f x x h L x h

0 x L

CPSTP

1 2

Similar 's are the same shape.

If you want more than that...

SohCahToaQED

CONCEPTUAL INSIGHT The examples in this section were selected because they lead to optimization problems where the min or max occurs at a critical point. Often, the critical point represents the best compromise between “competing factors.” In Example 3, we maximized profit by finding the best compromise between raising the rent and keeping the apartment units occupied. In Example 4, our solution minimizes surface area by finding the best compromise between height and width. In daily life, however, we often encounter endpoint rather than critical point solutions. For example, to run 10 meters in minimal time, you should run as fast as you can—the solution is not a critical point but rather an endpoint (your maximum speed).