Local and Absolute Maxima and Minima

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

A local extremum compares nearby inputs; an absolute extremum compares the whole stated domain. Domain endpoints and whether they are included can determine whether a largest or smallest value exists.

Method

An absolute minimum at $a$ satisfies $f(a)\le f(x)$ for every allowed $x$ . A local minimum needs this only in a neighborhood of $a$ within the domain. Maximum reverses the inequality.

Worked example

For $f(x)=(x-1)^2$ on $[0,3]$ , the absolute minimum is $0$ at $x=1$ . Endpoint values are $1$ and $4$ , so the absolute maximum is $4$ at $x=3$ . On the open interval $(0,3)$ , values approach $4$ but never attain it, so no absolute maximum exists.

Common mistake

A bound need not be attained. The words “maximum” and “minimum” require actual inputs that produce those values.

Check your understanding

Find the absolute extrema of $x^2$ on $[-2,1]$ .

Show answer

Minimum $0$ at $0$ ; maximum $4$ at $-2$ .

Explore

Move and scale a graph

Try this. Change one slider at a time. Positive h moves the vertex right; k moves it up; negative a reflects the curve and a = 0 makes it constant.

Vertical multiplier a1Horizontal shift h0Vertical shift k0

y = 1(x − (0))² + (0). Vertex (0, 0); opens up. Range y ≥ 0.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

For f(x)=x² on [−1,2], where is its absolute maximum?

x=−1

x=2

x=0

Hint 1 · Find a starting point

Compare all possible endpoint and turning-point values.

Hint 2 · Take the next step

The values at −1, 0 and 2 are 1, 0 and 4.

Show the reasoning

Answer: x=2

The largest attained value is 4 at x=2; x=0 is the minimum.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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