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Maxima and Minima

The derivative measures the slope of the tangent to a curve.At the maximum or minimum points, the tangent is horizontal and has slope zero.

Setting the derivative to zero, enables us to locate these max/min points.

This process tells us where these max/min points are but not whether they are a maximum or minimum.

So how can we tell ??

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x

y

x1 x2

The first derivative would give us the value of x1 and x2 but not the fact that x1 is a minimum point and x2 is a maximum point.We can establish this by taking the second derivative, i.e. calculating the derivative of the derivative.

0y

0y

3

Test :

If , at a turning point (max/min point) then the point is a minimum.

If , the point is maximum.

0y

0y

4

Example:

We know this has a maximum since

Setting to find any turning points :

turning point

Since -8 < 0 then this point is a maximum.

754 2 xxy

58 xy0y

580 x x85

85x

8y

04 a

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Example:

Turning points occur when

Since ln x is undefined when x < 0, then 1.581 is the only solution

xxy ln52

xxxxy 52152

0y

xxxx 25 520

25 25 22 xx

581.125 x

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When x = 1.581

Since 4 > 0, the point is a minimum.

xxy ln52 15252 xxxxy

Now, 252 xy

4581.152 2 y

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Note: If , the point might not be a maximum or minimum.

point of inflection

0y

0

0

y

y

0

0

y

y

0

0

y

y

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Solving Verbal Problems:

(1) Read the question carefully.

(2) Create an equation which gives a mathematical description of the process. (Sometimes given).

(3) Calculate the derivative and find turning points.(4) Establish whether or not turning points are maximum or minimum or neither.(5) Interpret and explain the results in sentences.

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Note : (3)

If using the derivative does not produce the desired result, then the result will be one of the end points.

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e.g. looking for the minimum value of some function

A B C

The first derivative will identify the maximum value (B) because it is a turning point.

We are searching for the min value so check the value of the function at A and C. The lesser of the two gives the desired result.

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Example:

The cost per hour C (in dollars) of operating an automobile is given by

600 ,08.00012.012.0 2 sssC

where s is the speed in miles per hour. At what speed is the cost per hour a minimum?

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Given

Setting gives

s = 50 is a turning point

:. The turning point is a maximum.

sC

ssC

0024.012.0

08.00012.012.0 2

500024.0

12.0

0024.012.00

s

s

0C

00024.0 C

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Check endpoints of s.

The minimum value occurs at s = 0.

Costs are minimised when speed = 0 mi/hr.

600 s

08.0

08.000012.0012.00 2

C

96.2

08.0600012.06012.060 2

C

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Example:

For a monopolist, the cost per unit of producing a product is $3 and the demand equation is

What price will give the greatest profit?

.10 qp

unitper 3$ 10 C

qp

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Profit = total revenue - total costs

35 21

qPqqP 310 21

qqP 310

qqq

P 3.10

qpqP 3

since 21

21

1

qq

q

16

Setting

35 21

qP

0P

350 21

q

21

53

q

22

12

53

q

925

778.2q

17

q = 2.778 is a turning point

:. q = 2.778 is a maximum point

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778.25.2 ,778.2When

Pq

23

5.2

qP

054.0

The price that generates the maximum profit is

6778.2

1010 q

p

A price of $6 for the product will produce maximum profit.

35 21

qP

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Example

The cost of operating a truck on a throughway (excluding the salary of the driver) is 0.11+(s/300) dollars per mile, where s is the (steady) speed of the truck in miles per hour.

The truck driver’s salary is $12 per hour.

At what speed should the truck driver operate the truck to make a 700-mile trip most economical?

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Cost per mile

---------(1)

:. For 700 miles,

Driver’s salary = $12/hr,

number of hours =s

700Since distance = speed time

:. Cost of driver ----------(2)ss

840070012

30011.0

s

30011.0700Cost

s

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Total cost = (1) + (2)

2840037 sC

ssC

840037

77

ss

C8400

30011.0700

21

Turning point60s

2

1

21

2

840037

s

8400372

s

37

8400 2 s

2840037

0 s

0 Setting C

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Checking with

:. The turning point is a minimum.

So costs are minimised when s = 60 mi/hr

316800 sC

0078.0

6016800 ,60 3

Cs

C

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A final note :

It is worthwhile checking end points of the variables domain even when the calculus has produced a result.

A BCalculus will reveal a maximum point at A but it can be seen the max value of the function occurs at B