Maximum and Minimum Values

Maximum and Minimum Values — Moosa Academy

Asking for the largest value a function reaches has two different answers depending on where you look. A peak may be the highest point in its neighbourhood while another peak elsewhere is higher still. That distinction — local against absolute extreme values — is the subject of this lesson.

Concept Local and absolute

Local maximum — higher than everything immediately around it.

Absolute maximum — the highest value on the whole interval.

The same two ideas apply, inverted, to minima.

Every absolute extremum is also a local one; the reverse is not true. A hill is a hill whether or not a taller mountain stands further along.

Concept Where extreme values occur

f'(x) = 0

At a peak the function stops rising and has not yet begun to fall; at a valley the reverse. In both cases the instantaneous rate of change is momentarily zero, and the tangent is horizontal. Points where $f'(x) = 0$ — or where the derivative does not exist — are called critical points.

Critical points are candidates, not guarantees. The derivative of $x^3$ is zero at the origin, yet the curve keeps rising straight through — that is an inflection point, not an extremum.

Note Finding the absolute extremes on a closed interval

A continuous function on a closed interval $[a,\ b]$ always attains both an absolute maximum and an absolute minimum. To find them:

Find every critical point inside the interval by solving $f'(x) = 0$ .
Evaluate $f$ at each of those points.
Evaluate $f$ at both endpoints $a$ and $b$ .
The largest of those values is the absolute maximum, the smallest the absolute minimum.

The endpoints must be checked. A function can reach its highest value at the very edge of the interval, where no derivative test would ever flag it.

Example A cubic on a closed interval

Find the absolute extreme values of $f(x) = x^3 - 3x$ on $[-2,\ 3]$ .

f'(x) = 3x^2 - 3 = 3(x^2 - 1)

Setting this to zero gives

x = -1

and

x = 1

, both inside the interval.

f(-1) = -1 + 3 = 2

— a local maximum

f(1) = 1 - 3 = -2

— a local minimum

f(-2) = -8 + 6 = -2

— endpoint

f(3) = 27 - 9 = 18

— endpoint

⟹ absolute maximum 18 at x = 3, absolute minimum −2 at x = 1 and x = −2

The absolute maximum sits at an endpoint, not at either critical point. Skipping step 3 would have given the wrong answer of 2.

Note Telling a maximum from a minimum

Look at the sign of the derivative on either side of a critical point:

Positive then negative — the function rose and then fell, so this is a maximum.

Negative then positive — it fell and then rose, so this is a minimum.

The same sign on both sides — neither; the curve merely flattened and continued.

Summary

A local extremum beats its immediate neighbourhood; an absolute one beats the entire interval.
Extreme values occur where f′(x) = 0 or the derivative fails to exist — the critical points.
On a closed interval, check every critical point and both endpoints.
A critical point is only a candidate: the sign change of f′ decides whether it is a maximum, a minimum or neither.