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Calculus IApplications

Maximum and minimum values, from peaks to proof.

Read peaks and valleys, find critical numbers, and compare every candidate. Then work through the questions from your notes with each answer checked.

Recognize extrema · Step 1 of 5

Local means nearby; absolute means everywhere

−3−2−10123−2−1012345Select point AASelect point BBSelect point CCSelect point DD
A maximum is a highest height; a minimum is a lowest height. Absolute compares every point in the domain. Local (or relative) compares only nearby points. Here local extrema use an open neighbourhood inside the domain; endpoints are still candidates for absolute extrema.
Try it: Pick a marked point on the graph or use the buttons. Compare it with its neighbours, then with the whole graph.
Point

B is a local maximum: it is higher than nearby points. Its height 3 is not the absolute maximum, because D is higher.

c=−1,f(c)=3{\displaystyle c} = {\displaystyle -1,\quad f(c)} = {\displaystyle 3}

The input c is the location, f(c) is the value, and (c, f(c)) is the point.

f(c)≥f(x)for a maximum{\displaystyle f(c)} \ge {\displaystyle f(x)\quad\text{for a maximum}}
f(c)≤f(x)for a minimum{\displaystyle f(c)} \le {\displaystyle f(x)\quad\text{for a minimum}}

Use all domain inputs for an absolute extremum, or just nearby inputs for a local one. Equal heights are allowed; extrema need not be unique.

Check yourself
Can a local maximum be lower than another point farther away?

Not sure which points to compare?

Bring your practice questions. We work through them together until every type feels routine.

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