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How to find local and absolute maxima and minima from a graph

The rule: local compares nearby points, absolute compares the whole graph

A local maximum is a peak relative to nearby points, and a local minimum is a nearby valley. An absolute maximum is the greatest included $y$ -value anywhere on the displayed domain; an absolute minimum is the least. Endpoints can be absolute extrema even when a course does not count them as local extrema. An open circle is not part of the function and cannot be an attained maximum or minimum.

Problem

A continuous graph of $y = f (x)$ starts at $(1, 7)$ , falls to $(2, 3)$ , rises to $(3, 4)$ , and then falls to the endpoint $(5, 1)$ . Find the local maximum, local minimum, absolute maximum and absolute minimum.

The interior turning points give the local extrema; the highest and lowest included points give the absolute extrema.

Answer

Local maximum:

f (3) = 4

. Local minimum:

f (2) = 3

. Absolute maximum:

f (1) = 7

. Absolute minimum:

f (5) = 1

All four values were checked against the complete stated graph and its included endpoints.

Step-by-step solution

Find the local extrema

Follow the graph from left to right. It falls until $(2, 3)$ and then rises, so the direction changes from decreasing to increasing there. That makes

f (2) = 3

a local minimum.

The graph then rises to $(3, 4)$ and falls afterward. The direction changes from increasing to decreasing, so

f (3) = 4

is a local maximum.

Find the absolute extrema

Now compare every height on the whole domain, including the endpoints. The highest included point is $(1, 7)$ , hence

f (1) = 7

is the absolute maximum. The lowest included point is $(5, 1)$ , hence

f (5) = 1

is the absolute minimum.

Why the answers differ

The local maximum value $4$ does not have to be the largest value overall; it only needs to be larger than nearby values. Likewise, the local minimum value $3$ is not the absolute minimum because the graph later reaches $1$ .

How to check yourself

Use direction changes for local extrema and compare all included heights for absolute extrema. Before choosing any marked point, check whether it is filled or open and whether the endpoint convention in the course allows endpoints to count as local extrema.

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