maximum/minimum 331

Maxima and minima

minimum (or relative minimum) for a graph is the

location of a function value smaller than all nearby

function values. Again, a local minimum need not be

an absolute minimum, although it could be.

If the function in question is differentiable, then

the tools and techniques of

CALCULUS

allow one to

readily locate local maxima and minima. It is clear

geometrically, for example, that the slope of the tan-

gent line to a graph is zero at a local maximum or

local minimum. We have:

If the function f(x) has a local maximum or

local minimum at x= c, then f′(c) = 0

The limit definition of the derivative provides a precise

proof of this. If the value f(c) is a local maximum, for

example, then, for small h, the value f(c+ h)is less than

the value f(c).Consequently, if happroaches the value

zero by running through positive values just above

zero, then the quotient is negative. This

shows that the derivative

must be ≤0. On the other hand, if happroaches zero

through negative values, then the quotient is positive,

and f′(c) ≥0. It must be the case then that f′(c) = 0.

Any value x= cfor which f′(c) = 0 is called a criti-

cal point (or a stationary point) for the function. A

study of

INCREASING

/

DECREASING

functions establishes:

A critical point x= cis a local maximum for

the function fif, and only if, f(x) is increasing

just to the left of cand decreasing just to the

right of c. Consequently, x= cis a local maxi-

mum if, and only if, f′(c) = 0 and f′(x) > 0 just

to the left of c, and f′(x) < 0 just to its right.

A critical point x= cis a local minimum

for the function fif, and only if, f(x) is decreas-

ing just to the left of cand increasing just to the

right of c. Consequently, x= cis a local mini-

mum if, and only if, f′(c) = 0 and f′(x) < 0 just

to the left of c, and f′(x) > 0 just to its right.

This observation, called the first-derivative test, allows

one to determine whether or not a critical point is

a local maximum or a local minimum. (Any critical

point at which the derivative does indeed change sign

is called a turning point.) As an example, consider

the function . To find its critical points

we need to solve the equation f′(x) = 0. This gives:

, yielding x= –1