Worksheets · Calculus

Absolute extrema on a closed interval worksheet

On a closed interval the largest value is either at a turning point or at one of the two ends, so the whole question is a short list of candidates and a comparison of their heights.

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Sigma Prep · sigmaprep.io/worksheets

Name

Absolute Extrema on a Closed Interval

Date Period

Find the absolute maximum and minimum of each function on the given interval.

  1. Find absolute extrema on a closed interval

    Find the absolute maximum and minimum of f(x)=4x3+24x2−60x+9 on [−6,3].
  2. Find the absolute maximum and minimum of f(x)=2x3+9x2+12x−1 on [−3,1].
  3. Find the absolute maximum and minimum of f(x)=4x3+30x2+72x+5 on [−4,1].
  4. Find the absolute maximum and minimum of f(x)=4xx2+1 on [−2,1].
  5. Find the absolute maximum and minimum of f(x)=2x3+6x2−48x−4 on [−5,4].
  6. Find the absolute maximum and minimum of f(x)=x3−3x+2 on [−3,3].
  7. Find the absolute maximum and minimum of f(x)=2x3−27x2+120x−9 on [1,6].
  8. Find the absolute maximum and minimum of f(x)=(x+3)23 on [−4,5].
  9. Find the absolute maximum and minimum of f(x)=−2x3+27x2−120x−9 on [1,6].
  10. Find the absolute maximum and minimum of f(x)=2x3+3x2+2 on [−3,0].
  11. Find the absolute maximum and minimum of f(x)=−4x3−48x2−180x−3 on [−7,0].
  12. Find the absolute maximum and minimum of f(x)=x4−8x2 on [−4,5].
  13. Find the absolute maximum and minimum of f(x)=−2x3+9x2+24x+8 on [−4,5].

  1. Find absolute extrema of non-polynomials

    On [−4,2], find where f(x)=xx2+25 takes its absolute maximum and minimum, and give both values.
  2. Find the largest and smallest values of f(x)=sin⁡x+cos⁡x on [0,π], and where they occur.
  3. Find the absolute maximum and minimum of f(x)=x−6x on [0,81].
  4. Find the largest and smallest values of f(x)=x+25x on [1,8], and where they occur.
  5. Give the absolute extrema of f(x)=xe−x on the closed interval [−1,3].
  6. Find the absolute maximum and minimum of f(x)=xe−x2 on [0,4].
  7. Give the absolute extrema of f(x)=x−2x on the closed interval [0,16].

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Absolute Extrema on a Closed IntervalAnswer keyVersion 1

Date Period

Find the absolute maximum and minimum of each function on the given interval.

  1. Find absolute extrema on a closed interval

    Find the absolute maximum and minimum of f(x)=4x3+24x2−60x+9 on [−6,3].max 409 at x=−5, min −23 at x=1
  2. Find the absolute maximum and minimum of f(x)=2x3+9x2+12x−1 on [−3,1].Max: 22 at x=1Min: −10 at x=−3
  3. Find the absolute maximum and minimum of f(x)=4x3+30x2+72x+5 on [−4,1].max 111 at x=1, min −59 at x=−4
  4. Find the absolute maximum and minimum of f(x)=4xx2+1 on [−2,1].Max: 2 at x=1Min: −2 at x=−1
  5. Find the absolute maximum and minimum of f(x)=2x3+6x2−48x−4 on [−5,4].max 156 at x=−4, min −60 at x=2
  6. Find the absolute maximum and minimum of f(x)=x3−3x+2 on [−3,3].Max: 20 at x=3Min: −16 at x=−3
  7. Find the absolute maximum and minimum of f(x)=2x3−27x2+120x−9 on [1,6].max 171 at x=6, min 86 at x=1
  8. Find the absolute maximum and minimum of f(x)=(x+3)23 on [−4,5].Max: 4 at x=5Min: 0 at x=−3
  9. Find the absolute maximum and minimum of f(x)=−2x3+27x2−120x−9 on [1,6].max −104 at x=1, min −189 at x=6
  10. Find the absolute maximum and minimum of f(x)=2x3+3x2+2 on [−3,0].Max: 3 at x=−1Min: −25 at x=−3
  11. Find the absolute maximum and minimum of f(x)=−4x3−48x2−180x−3 on [−7,0].max 277 at x=−7, min −3 at x=0
  12. Find the absolute maximum and minimum of f(x)=x4−8x2 on [−4,5].Max: 425 at x=5Min: −16 at x=−2 and x=2
  13. Find the absolute maximum and minimum of f(x)=−2x3+9x2+24x+8 on [−4,5].max 184 at x=−4, min −5 at x=−1

  1. Find absolute extrema of non-polynomials

    On [−4,2], find where f(x)=xx2+25 takes its absolute maximum and minimum, and give both values.max 229 at x=2, min −441 at x=−4
  2. Find the largest and smallest values of f(x)=sin⁡x+cos⁡x on [0,π], and where they occur.max 2 at x=π4, min −1 at x=π
  3. Find the absolute maximum and minimum of f(x)=x−6x on [0,81].max 27 at x=81, min −9 at x=9
  4. Find the largest and smallest values of f(x)=x+25x on [1,8], and where they occur.max 26 at x=1, min 10 at x=5
  5. Give the absolute extrema of f(x)=xe−x on the closed interval [−1,3].max 1e at x=1, min −e at x=−1
  6. Find the absolute maximum and minimum of f(x)=xe−x2 on [0,4].max 2e at x=2, min 0 at x=0
  7. Give the absolute extrema of f(x)=x−2x on the closed interval [0,16].max 8 at x=16, min −1 at x=1

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the absolute maximum and minimum of y=x3−3x on [0,3]

  1. The derivative is 3x squared minus 3, which is zero at x equals 1 and x equals negative 1.
  2. Only x equals 1 is inside the interval, so the candidates are 0, 1 and 3.
  3. The heights there are 0, negative 2 and 18, so the maximum is 18 and the minimum is negative 2.

The answer is max 18 at x=3, min −2 at x=1.

Where students go wrong

Using a turning point that falls outside the interval. Only the critical points between the two ends count, and once one is dropped the largest value is often at an end instead.

Also called absolute maximum, absolute minimum, global extrema, extreme value theorem, candidates test or closed interval method.

What you can put on this worksheet

Absolute extrema on a closed interval
Find the absolute maximum and minimum of f(x)=x2+4x+7 on [−4,1]. → max 12 at x=1, min 3 at x=−2
Trig, exponential, log and rational
Give the absolute extrema of f(x)=xx2+9 on the closed interval [0,4]. → max 16 at x=3, min 0 at x=0
Quartics, cusps, cubics and rational functions
Find the absolute maximum and minimum of f(x)=(x+3)2+3 on [−5,0]. → Max: 12 at x=0Min: 3 at x=−3

Questions about these worksheets

Yes. Every absolute extrema on a closed interval sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the absolute extrema on a closed interval questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: absolute extrema on a closed interval, trig, exponential, log and rational and quartics, cusps, cubics and rational functions. Tick as many as you want and set how many of each, or let it spread them evenly.

Using a turning point that falls outside the interval. Only the critical points between the two ends count, and once one is dropped the largest value is often at an end instead.