and x= 1 as critical points. For a value just to the left of

x= –1 (say, x = –1.1), we see that f′(x) is negative. For a

value just to the right of x= –1 (say, x= –0.9), it is posi-

tive. Thus the function has a local minimum at x= –1.

Similar analysis shows that x= 1 is a local maximum.

A study of

CONCAVE UP

/

CONCAVE DOWN

functions

establishes:

At a local maximum, a function is concave

down. Consequently, a local maximum for a

function f(x) can be identified as a point x= c

with f′(c) = 0 and f′′(c) < 0.

At a local minimum, a function is concave

up. Consequently, a local minimum for a func-

tion f(x) can be identified as a point x= cwith

f′(c) = 0 and f′′(c) > 0.

This observation is called the second-derivative test.

For example, consider the challenge of finding two

positive numbers whose sum is 100 and whose product

is as large as possible. If we let xand yrepresent two

variable positive numbers, then we are being asked to

“maximize” the product function p= xy, given that

x+y= 100. Solving for yand substituting yields: P=

x(100 – x) = 100x– x2. This expresses Pas a function

of xalone. A maximum for this function can only

occur at a critical point. Solving = 0 yields 100 – 2x

= 0 or x= 50. To classify this critical point, note that

= –2 < 0. This shows that the curve is concave

down. Consequently, the point x= 50, the only critical

point, is a (global) maximum. The corresponding value

for yis y= 100 – 50 = 50. The maximum value of the

product is thus 50 ×50 = 2,500.

See also

OPTIMIZATION

.

Mayan mathematics The Mayan culture, based in

the Yucatán Peninsula, Central America, spanned the

period of 250

C

.

E

. to 900

C

.

E

., but was based on a civi-

lization established in the region as early as 2000

B

.

C

.

E

.

The people of this time had built large cities, replete

with construction of all types, including large reservoirs

for storing rainwater, sophisticated irrigation systems,

and raised fields for farming. The rulers were priests,

and religious practices were synchronized with astro-

nomical observations

In 1511, with a force of 11 ships and 508 soldiers,

Spanish explorer Hernán Cortés conquered the peoples

of the Yucatán. He met next to no resistance. Thirty

years later, Bishop Diego de Landa of the Franciscan

Order was sent to the New World as a missionary, and

he tried his best to help the Maya peoples and protect

them from their new Spanish masters. At first he was

horrified by the religious practices of the people and

the unusual icons that appeared in their written texts,

believing the writing to be the lies of the devil. He

ordered that all Mayan idols be destroyed and all the

Mayan books be burned, which grieved the local peo-

ple considerably. Perhaps out of remorse, Landa felt the

need to write in 1566 a text, Relación de las cosas de

Yucatán (About things of the Yucatán), recording the

writing, customs, religious practices, and history of the

Maya people, along with some attempt at justifying his

actions. It is from this work, and four remnants of

actual Mayan text saved from the fire, that we learn of

Mayan mathematics. Much of the mathematics of the

people was motivated by computations of the planet

cycles and a need to keep an accurate calendar system.

Coming from a tropical climate, the Maya were

fully aware that the human body comes with 20 digits,

and it is not surprising that these people developed a

base-20 system of counting (today called a vigesimal

system). Thus, instead of counting in groups of units

(1), tens (10), one hundreds (100), and one thousands

(1,000), people of this culture counted in terms of units

(1) and twenties (20). Motivated by the number of days

in the year, the Maya next worked in groups of 360,

and then groups 20 ×360. Thus, for example, a num-

ber depicted as 2-12-5-17 in the Mayan system repre-

sents the value:

17 ×1 + 5 ×20 + 12 ×360 + 2 ×20 ×360 = 18,837

(The system is thus not strictly vigesimal.)

Numbers were represented as dots and bars, with a

bar denoting five units. Thus a diagram depicting a

group of two bars and three dots represents the number

13. (Some historians suggest that each dot represents a

finger tip, and the horizontal bar an outstretched hand.)

The Maya used positional notation to represent large

numbers, including a symbol for zero, the picture of a

closed fist, or perhaps a conch shell, to represent no dig-

its of a certain power of 20 or 360.

The Maya had two calendar systems. The Tzolkin,

the ritual calendar, contained 13 months, one for each

god of the upper world, with 20 days per month. Thus

d2P

––

dx2

dP

––

dx

332 Mayan mathematics