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Absolute Extrema worksheets

Find exact absolute minimum and maximum values of quadratic, cubic, and quartic polynomials on the real line or a stated closed interval.

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Read the stated domain, find every in-domain stationary point, compare exact function values, and use end behavior when the domain is all real numbers.

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Global answers depend on the domain

A closed interval requires both endpoints and every stationary point inside it. On all real numbers, polynomial end behavior decides whether an absolute minimum or maximum can exist.

Learning objective

What this activity builds

Compare every required candidate exactly and report each absolute extremum value with all inputs where it is attained.

Worked example

Find the absolute extrema of f(x) = x^2 on [-1, 2].

Absolute minimum 0 at x = 0; absolute maximum 4 at x = 2.

The candidates are both endpoints and the stationary point: f(-1) = 1, f(0) = 0, and f(2) = 4.

Teaching notes

Make the practice count

  • Write both the value and every input where it occurs.
  • On a closed interval, always include both endpoints.
  • On the real line, decide boundedness from polynomial end behavior.

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Math SheetNameScoreTeacherDateAbsolute ExtremaFind the absolute minimum and maximum values on the stated domain, including every input where each value is attained. Explain when an extremum does not exist.1. Find the absolute minimum and maximum values and all locations wherethey occur. Explain if either does not exist.Domain: R (all real x)f(x) =x2+4·x+72. Find the absolute minimum and maximum values and all locations wherethey occur. Explain if either does not exist.Domain: R (all real x)f(x) =x3+3·x2-23. Find the absolute minimum and maximum values and all locations wherethey occur. Explain if either does not exist.Domain: R (all real x)f(x) =2·x4+8·x3-16·x-134. Find the absolute minimum and maximum values and all locations wherethey occur. Explain if either does not exist.Domain: [-2, 2]f(x) =x2-2·x+5Compare endpoints and every in-domain stationary point. Use end behavior on R.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateAbsolute ExtremaFind the absolute minimum and maximum values on the stated domain, including every input where each value is attained. Explain when an extremum does not exist.5. Find the absolute minimum and maximum values and all locations wherethey occur. Explain if either does not exist.Domain: [-6, 6]f(x) =-x3+3·x2-x-56. Find the absolute minimum and maximum values and all locations wherethey occur. Explain if either does not exist.Domain: [-4, 4]f(x) =-2·x4+8·x2-17. Find the absolute minimum and maximum values and all locations wherethey occur. Explain if either does not exist.Domain: R (all real x)f(x) =-x2+28. Find the absolute minimum and maximum values and all locations wherethey occur. Explain if either does not exist.Domain: R (all real x)f(x) =-x3+2Compare endpoints and every in-domain stationary point. Use end behavior on R.math-sheet.comPage 2 of 2

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