Worksheets · Calculus

Concavity and inflection points worksheet

The second derivative does for the bend what the first does for the slope: positive means the curve holds water, negative means it sheds it, and a sign change is an inflection point.

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Name

Concavity and Inflection Points

Date Period

Answer each question about the concavity of the curve.

  1. Find intervals of concavity

    y=3x4−30x3−108x2−1. Where is y concave up?
  2. y=−x4−8x3+30x2+9x−9. Where is y concave up?
  3. y=−x4−2x3+180x2−3x−8. Where is y concave up?
  4. y=−3x4+6x3+216x2+5x−2. Where is y concave down?
  5. y=2x4+8x3−36x2+x−5. Where is y concave down?

  1. Find concavity and inflection from f′f'

    The graph of f′, the derivative of f, is shown. On what open intervals is f concave up?−4−3−2−101234−3−2−10123f′
  2. The graph of f′, the derivative of f, is shown. At what x does f have a point of inflection?−4−3−2−101234−3−2−10123f′
  3. The graph of f′, the derivative of f, is shown. On what open intervals is f concave down?−3−2−101234−3−2−10123f′
  4. The graph of f′, the derivative of f, is shown. On what open intervals is f concave down?−1012345678−3−2−10123f′
  5. The graph of f′, the derivative of f, is shown. On what open intervals is f concave up?−2−10123456−3−2−10123f′

  1. Find inflection points of non-polynomials

    Find the x-values of the inflection points of f(x)=sin⁡2x on (0,2π)
  2. Find the x-values of the inflection points of f(x)=sin⁡x+cos⁡x on (0,2π)
  3. Find the x-values of the inflection points of f(x)=x2ex
  4. Find the x-values of the inflection points of f(x)=e−x2
  5. Find the x-values of the inflection points of f(x)=x2ln⁡x

  1. Find concavity and inflection points

    Find the intervals on which f(x)=6x−1 is concave up and concave down.
  2. Find the intervals on which f(x)=x3−12x2−6 is concave up and concave down, and the inflection points (x-values).
  3. Find the intervals on which f(x)=xex is concave up and concave down, and the inflection points (x-values).
  4. Find the intervals on which f(x)=−(x+3)ex is concave up and concave down, and the inflection points (x-values).
  5. Find the intervals on which f(x)=−2x3+3x+6 is concave up and concave down, and the inflection points (x-values).

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

Sigma Prep · sigmaprep.io/worksheets

Name

Concavity and Inflection PointsAnswer keyVersion 1

Date Period

Answer each question about the concavity of the curve.

  1. Find intervals of concavity

    y=3x4−30x3−108x2−1. Where is y concave up?(−∞,−1)∪(6,∞)
  2. y=−x4−8x3+30x2+9x−9. Where is y concave up?(−5,1)
  3. y=−x4−2x3+180x2−3x−8. Where is y concave up?(−6,5)
  4. y=−3x4+6x3+216x2+5x−2. Where is y concave down?(−∞,−3)∪(4,∞)
  5. y=2x4+8x3−36x2+x−5. Where is y concave down?(−3,1)

  1. Find concavity and inflection from f′f'

    The graph of f′, the derivative of f, is shown. On what open intervals is f concave up?−4−3−2−101234−3−2−10123f′(−4,−3)∪(−2,3)
  2. The graph of f′, the derivative of f, is shown. At what x does f have a point of inflection?−4−3−2−101234−3−2−10123f′x=−3, x=1, and x=3
  3. The graph of f′, the derivative of f, is shown. On what open intervals is f concave down?−3−2−101234−3−2−10123f′None
  4. The graph of f′, the derivative of f, is shown. On what open intervals is f concave down?−1012345678−3−2−10123f′(−1,3)∪(6,7)
  5. The graph of f′, the derivative of f, is shown. On what open intervals is f concave up?−2−10123456−3−2−10123f′(1,5)

  1. Find inflection points of non-polynomials

    Find the x-values of the inflection points of f(x)=sin⁡2x on (0,2π)x=π4, x=3π4, x=5π4, x=7π4
  2. Find the x-values of the inflection points of f(x)=sin⁡x+cos⁡x on (0,2π)x=3π4,x=7π4
  3. Find the x-values of the inflection points of f(x)=x2exx=−2−2, x=−2+2
  4. Find the x-values of the inflection points of f(x)=e−x2x=−22,x=22
  5. Find the x-values of the inflection points of f(x)=x2ln⁡xx=e−32

  1. Find concavity and inflection points

    Find the intervals on which f(x)=6x−1 is concave up and concave down.Concave up: (1,∞)Concave down: (−∞,1)
  2. Find the intervals on which f(x)=x3−12x2−6 is concave up and concave down, and the inflection points (x-values).Concave up: (4,∞)Concave down: (−∞,4)Inflection at x=4
  3. Find the intervals on which f(x)=xex is concave up and concave down, and the inflection points (x-values).Concave up: (−2,∞)Concave down: (−∞,−2)Inflection at x=−2
  4. Find the intervals on which f(x)=−(x+3)ex is concave up and concave down, and the inflection points (x-values).Concave up: (−∞,−5)Concave down: (−5,∞)Inflection at x=−5
  5. Find the intervals on which f(x)=−2x3+3x+6 is concave up and concave down, and the inflection points (x-values).Concave up: (−∞,0)Concave down: (0,∞)Inflection at x=0

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

How to do these

Where is y=x3−6x2+4 concave up?

  1. The first derivative is 3x squared minus 12x.
  2. The second derivative is 6x minus 12, which is zero at x equals 2.
  3. It is positive to the right of 2, so the curve is concave up on 2 to infinity.

The answer is (2,∞).

Where students go wrong

Calling every zero of the second derivative an inflection point. It is only one if the bend actually changes, so the sign has to be different on the two sides, not merely zero in the middle.

Also called intervals of concavity, concave up, concave down, inflection point, points of inflection or second derivative test for concavity.

What you can put on this worksheet

Intervals of concavity
y=2x3+30x2+x. Where is y concave down? → (−∞,−5)
Inflection points
Find the x-coordinate of the inflection point of y=8x3−72x2−8 → x=3
Concavity from the graph of f'
The graph of f′, the derivative of f, is shown. At what x does f have a point of inflection? — the graph of f′ on [-1, 8] → x=5
Non-polynomial functions
Find the x-values of the inflection points of f(x)=1x2+3 → x=−1,x=1
Across an asymptote and around a cusp
Find the intervals on which f(x)=x−13 is concave up and concave down, and the inflection points (x-values). → Concave up: (−∞,1)Concave down: (1,∞)Inflection at x=1

Questions about these worksheets

Yes. Every concavity and inflection points 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 concavity and inflection points 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 5: intervals of concavity, inflection points, concavity from the graph of f', non-polynomial functions and across an asymptote and around a cusp. Tick as many as you want and set how many of each, or let it spread them evenly.

Calling every zero of the second derivative an inflection point. It is only one if the bend actually changes, so the sign has to be different on the two sides, not merely zero in the middle.