Worksheets · Calculus

Relative extrema worksheet

Finding the critical points is half the job; deciding which is a peak and which is a valley is the other half. The first derivative test reads the sign either side, and the second derivative test reads the bend.

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Name

Relative Extrema

Date Period

Find the relative extrema of each function. Give each as a point and state whether it is a maximum or a minimum.

  1. Find relative extrema

    Find the relative extrema of y=4x3+6x2−72x+9
  2. Find the relative extrema of f(x)=x3+2x2+x−4.
  3. Find the relative extrema of y=6x3+72x2+126x−4
  4. Find the relative extrema of f(x)=6xx2+16.
  5. Find the relative extrema of y=−2x3+96x−2
  6. Find the relative extrema of f(x)=x2+16x.
  7. Find the relative extrema of y=2x3−33x2+180x−9
  8. Find the relative extrema of f(x)=x3−4x2−3x−5.
  9. Find the relative extrema of y=−2x3+6x2+90x+7
  10. Find the relative extrema of f(x)=xx2+1.

  1. Use the second derivative test

    Use the second derivative test to find the relative extrema of f(x)=x3−3x
  2. Use the second derivative test to find the relative extrema of f(x)=−x3−3x2+9x+5
  3. Use the second derivative test to find the relative extrema of f(x)=−x3−9x2−15x−5
  4. Use the second derivative test to find the relative extrema of f(x)=−x3+12x2−45x
  5. Use the second derivative test to find the relative extrema of f(x)=x3−3x2−7

  1. Find relative extrema of non-polynomials

    Find the relative extrema of f(x)=ln⁡xx.
  2. Find the relative extrema of f(x)=exx.
  3. Find the relative extrema of f(x)=xx.
  4. Find the relative extrema of f(x)=xx+1.
  5. Find the relative extrema of f(x)=exx−1.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Relative ExtremaAnswer keyVersion 1

Date Period

Find the relative extrema of each function. Give each as a point and state whether it is a maximum or a minimum.

  1. Find relative extrema

    Find the relative extrema of y=4x3+6x2−72x+9max (−3,171), min (2,−79)
  2. Find the relative extrema of f(x)=x3+2x2+x−4.max (−1,−4), min (−13,−11227)
  3. Find the relative extrema of y=6x3+72x2+126x−4max (−7,584), min (−1,−64)
  4. Find the relative extrema of f(x)=6xx2+16.min (−4,−34), max (4,34)
  5. Find the relative extrema of y=−2x3+96x−2min (−4,−258), max (4,254)
  6. Find the relative extrema of f(x)=x2+16x.max (−4,−8), min (4,8)
  7. Find the relative extrema of y=2x3−33x2+180x−9max (5,316), min (6,315)
  8. Find the relative extrema of f(x)=x3−4x2−3x−5.max (−13,−12127), min (3,−23)
  9. Find the relative extrema of y=−2x3+6x2+90x+7min (−3,−155), max (5,357)
  10. Find the relative extrema of f(x)=xx2+1.min (−1,−12), max (1,12)

  1. Use the second derivative test

    Use the second derivative test to find the relative extrema of f(x)=x3−3xmax (−1,2), min (1,−2)
  2. Use the second derivative test to find the relative extrema of f(x)=−x3−3x2+9x+5min (−3,−22), max (1,10)
  3. Use the second derivative test to find the relative extrema of f(x)=−x3−9x2−15x−5min (−5,−30), max (−1,2)
  4. Use the second derivative test to find the relative extrema of f(x)=−x3+12x2−45xmin (3,−54), max (5,−50)
  5. Use the second derivative test to find the relative extrema of f(x)=x3−3x2−7max (0,−7), min (2,−11)

  1. Find relative extrema of non-polynomials

    Find the relative extrema of f(x)=ln⁡xx.max (e,1e)
  2. Find the relative extrema of f(x)=exx.min (1,e)
  3. Find the relative extrema of f(x)=xx.min (1e,e−1e)
  4. Find the relative extrema of f(x)=xx+1.max (1,12)
  5. Find the relative extrema of f(x)=exx−1.min (2,e2)

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

How to do these

Find the relative extrema of y=x3−12x+5

  1. The derivative is 3x squared minus 12, which is zero at x equals 2 and x equals negative 2.
  2. The second derivative is 6x. At x equals negative 2 it is negative, so the curve is bending down and that is a maximum.
  3. At x equals 2 the second derivative is positive, so that one is a minimum.

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

Where students go wrong

Assuming every critical point is a maximum or a minimum. Where the derivative touches zero without crossing it the curve flattens and carries on the same way, so the sign either side has to be checked rather than assumed.

Also called first derivative test, second derivative test, local maximum, local minimum, relative maxima and minima or turning points.

What you can put on this worksheet

Relative maxima and minima
Find the relative extrema of y=6x2−8 → min (0,−8)
Critical points
Find the critical points of y=2x2−8 → x=0
Relative extrema from a graph
Name the relative maximum point and the relative minimum point of the function graphed. → Max (1,0), min (5,−5)
Read f from the graph of f'
The graph of f′, the derivative of f, is shown. At what x does f have a relative minimum on (−1,8)? — the graph of f′ on [-1, 8] → x=2
The second derivative test
Use the second derivative test to find the relative extrema of f(x)=x3−6x2−8 → max (0,−8), min (4,−40)
As points, for non-polynomials
Find the relative extrema of f(x)=x−3ln⁡x. → min (3,3−3ln⁡3)
Relative extrema as points
Find the relative extrema of f(x)=x3−9x2−5. → max (0,−5), min (6,−113)

Questions about these worksheets

Yes. Every relative extrema 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 relative extrema 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 7: relative maxima and minima, critical points, relative extrema from a graph, read f from the graph of f', the second derivative test, as points, for non-polynomials and relative extrema as points. Tick as many as you want and set how many of each, or let it spread them evenly.

Assuming every critical point is a maximum or a minimum. Where the derivative touches zero without crossing it the curve flattens and carries on the same way, so the sign either side has to be checked rather than assumed.