Find the critical points of f(x) = x + cos 2x on the open interval (0, 2π)

x = π/12, 5π/12, 13π/12, 17π/12

Critical points worksheet

A critical point is an x value where the derivative is zero or undefined. Every relative maximum and minimum is one, which is why this is the first step of almost every graphing question.

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Sigma Prep · sigmaprep.io/worksheets Name Critical Points Date Period Find the critical points of each function. Find critical points Find the critical points of $y = 6 x^{3} + 45 x^{2} - 252 x + 9$ Find the critical points of $y = - 2 x^{3} + 18 x^{2} + 42 x - 2$ Find the critical points of $y = 2 x^{3} - 6 x - 6$ Find the critical points of $y = 2 x^{3} - 9 x^{2} - 24 x + 5$ Find the critical points of $y = - 2 x^{3} - 15 x^{2} - 36 x + 8$ Find the critical points of $y = - 6 x^{3} - 72 x^{2} - 216 x - 2$ Find the critical points of $y = 2 x^{3} - 12 x^{2} - 72 x - 1$ Find critical points, including undefined $y'$ Find the critical points of $y = x^{\frac{2}{3}} (x - 15)$ Find the critical points of $y = x^{\frac{2}{3}} (x + 10)$ Find the critical points of $y = x + \frac{1}{x}$ Find the critical points of $y = x^{\frac{1}{3}} (x + 4)$ Find the critical points of $y = x^{\frac{2}{3}} (x - 5)$ Find the critical points of $y = x^{\frac{1}{3}} (x + 12)$ Find the critical points of $y = x^{\frac{1}{3}} (x + 8)$ Find critical points on $(0, 2\pi)$ Find the critical points of $f (x) = \sin 2 x + \cos 2 x$ on the open interval $(0, 2 π)$ Find the critical points of $f (x) = e^{x} \cos x$ on the open interval $(0, 2 π)$ Find the critical points of $f (x) = \sqrt{3} \sin \frac{x}{2} - \cos \frac{x}{2}$ on the open interval $(0, 2 π)$ Find the critical points of $f (x) = x + \cos 2 x$ on the open interval $(0, 2 π)$ Find the critical points of $f (x) = \sqrt{3} \sin 2 x - \cos 2 x$ on the open interval $(0, 2 π)$ Find the critical points of $f (x) = \frac{\cos x}{2 + \sin x}$ on the open interval $(0, 2 π)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Critical Points

Date Period

Find the critical points of each function.

Find critical points
Find the critical points of $y = 6 x^{3} + 45 x^{2} - 252 x + 9$
Find the critical points of $y = - 2 x^{3} + 18 x^{2} + 42 x - 2$
Find the critical points of $y = 2 x^{3} - 6 x - 6$
Find the critical points of $y = 2 x^{3} - 9 x^{2} - 24 x + 5$
Find the critical points of $y = - 2 x^{3} - 15 x^{2} - 36 x + 8$
Find the critical points of $y = - 6 x^{3} - 72 x^{2} - 216 x - 2$
Find the critical points of $y = 2 x^{3} - 12 x^{2} - 72 x - 1$

Find critical points, including undefined $y'$
Find the critical points of $y = x^{\frac{2}{3}} (x - 15)$
Find the critical points of $y = x^{\frac{2}{3}} (x + 10)$
Find the critical points of $y = x + \frac{1}{x}$
Find the critical points of $y = x^{\frac{1}{3}} (x + 4)$
Find the critical points of $y = x^{\frac{2}{3}} (x - 5)$
Find the critical points of $y = x^{\frac{1}{3}} (x + 12)$
Find the critical points of $y = x^{\frac{1}{3}} (x + 8)$

Find critical points on $(0, 2\pi)$
Find the critical points of $f (x) = \sin 2 x + \cos 2 x$ on the open interval $(0, 2 π)$
Find the critical points of $f (x) = e^{x} \cos x$ on the open interval $(0, 2 π)$
Find the critical points of $f (x) = \sqrt{3} \sin \frac{x}{2} - \cos \frac{x}{2}$ on the open interval $(0, 2 π)$
Find the critical points of $f (x) = x + \cos 2 x$ on the open interval $(0, 2 π)$
Find the critical points of $f (x) = \sqrt{3} \sin 2 x - \cos 2 x$ on the open interval $(0, 2 π)$
Find the critical points of $f (x) = \frac{\cos x}{2 + \sin x}$ on the open interval $(0, 2 π)$



Sigma Prep · sigmaprep.io/worksheets Name Critical PointsAnswer keyVersion 1 Date Period Find the critical points of each function. Find critical points Find the critical points of $y = 6 x^{3} + 45 x^{2} - 252 x + 9$ $x = - 7, x = 2$ Find the critical points of $y = - 2 x^{3} + 18 x^{2} + 42 x - 2$ $x = - 1, x = 7$ Find the critical points of $y = 2 x^{3} - 6 x - 6$ $x = - 1, x = 1$ Find the critical points of $y = 2 x^{3} - 9 x^{2} - 24 x + 5$ $x = - 1, x = 4$ Find the critical points of $y = - 2 x^{3} - 15 x^{2} - 36 x + 8$ $x = - 3, x = - 2$ Find the critical points of $y = - 6 x^{3} - 72 x^{2} - 216 x - 2$ $x = - 6, x = - 2$ Find the critical points of $y = 2 x^{3} - 12 x^{2} - 72 x - 1$ $x = - 2, x = 6$ Find critical points, including undefined $y'$ Find the critical points of $y = x^{\frac{2}{3}} (x - 15)$ $x = 0$ , $x = 6$ Find the critical points of $y = x^{\frac{2}{3}} (x + 10)$ $x = - 4$ , $x = 0$ Find the critical points of $y = x + \frac{1}{x}$ $x = - 1$ , $x = 1$ Find the critical points of $y = x^{\frac{1}{3}} (x + 4)$ $x = - 1$ , $x = 0$ Find the critical points of $y = x^{\frac{2}{3}} (x - 5)$ $x = 0$ , $x = 2$ Find the critical points of $y = x^{\frac{1}{3}} (x + 12)$ $x = - 3$ , $x = 0$ Find the critical points of $y = x^{\frac{1}{3}} (x + 8)$ $x = - 2$ , $x = 0$ Find critical points on $(0, 2\pi)$ Find the critical points of $f (x) = \sin 2 x + \cos 2 x$ on the open interval $(0, 2 π)$ $x = \frac{π}{8}, \frac{5 π}{8}, \frac{9 π}{8}, \frac{13 π}{8}$ Find the critical points of $f (x) = e^{x} \cos x$ on the open interval $(0, 2 π)$ $x = \frac{π}{4}, \frac{5 π}{4}$ Find the critical points of $f (x) = \sqrt{3} \sin \frac{x}{2} - \cos \frac{x}{2}$ on the open interval $(0, 2 π)$ $x = \frac{4 π}{3}$ Find the critical points of $f (x) = x + \cos 2 x$ on the open interval $(0, 2 π)$ $x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}$ Find the critical points of $f (x) = \sqrt{3} \sin 2 x - \cos 2 x$ on the open interval $(0, 2 π)$ $x = \frac{π}{3}, \frac{5 π}{6}, \frac{4 π}{3}, \frac{11 π}{6}$ Find the critical points of $f (x) = \frac{\cos x}{2 + \sin x}$ on the open interval $(0, 2 π)$ $x = \frac{7 π}{6}, \frac{11 π}{6}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Critical PointsAnswer keyVersion 1

Date Period

Find the critical points of each function.

Find critical points
Find the critical points of $y = 6 x^{3} + 45 x^{2} - 252 x + 9$ $x = - 7, x = 2$
Find the critical points of $y = - 2 x^{3} + 18 x^{2} + 42 x - 2$ $x = - 1, x = 7$
Find the critical points of $y = 2 x^{3} - 6 x - 6$ $x = - 1, x = 1$
Find the critical points of $y = 2 x^{3} - 9 x^{2} - 24 x + 5$ $x = - 1, x = 4$
Find the critical points of $y = - 2 x^{3} - 15 x^{2} - 36 x + 8$ $x = - 3, x = - 2$
Find the critical points of $y = - 6 x^{3} - 72 x^{2} - 216 x - 2$ $x = - 6, x = - 2$
Find the critical points of $y = 2 x^{3} - 12 x^{2} - 72 x - 1$ $x = - 2, x = 6$

Find critical points, including undefined $y'$
Find the critical points of $y = x^{\frac{2}{3}} (x - 15)$ $x = 0$ , $x = 6$
Find the critical points of $y = x^{\frac{2}{3}} (x + 10)$ $x = - 4$ , $x = 0$
Find the critical points of $y = x + \frac{1}{x}$ $x = - 1$ , $x = 1$
Find the critical points of $y = x^{\frac{1}{3}} (x + 4)$ $x = - 1$ , $x = 0$
Find the critical points of $y = x^{\frac{2}{3}} (x - 5)$ $x = 0$ , $x = 2$
Find the critical points of $y = x^{\frac{1}{3}} (x + 12)$ $x = - 3$ , $x = 0$
Find the critical points of $y = x^{\frac{1}{3}} (x + 8)$ $x = - 2$ , $x = 0$

Find critical points on $(0, 2\pi)$
Find the critical points of $f (x) = \sin 2 x + \cos 2 x$ on the open interval $(0, 2 π)$ $x = \frac{π}{8}, \frac{5 π}{8}, \frac{9 π}{8}, \frac{13 π}{8}$
Find the critical points of $f (x) = e^{x} \cos x$ on the open interval $(0, 2 π)$ $x = \frac{π}{4}, \frac{5 π}{4}$
Find the critical points of $f (x) = \sqrt{3} \sin \frac{x}{2} - \cos \frac{x}{2}$ on the open interval $(0, 2 π)$ $x = \frac{4 π}{3}$
Find the critical points of $f (x) = x + \cos 2 x$ on the open interval $(0, 2 π)$ $x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}$
Find the critical points of $f (x) = \sqrt{3} \sin 2 x - \cos 2 x$ on the open interval $(0, 2 π)$ $x = \frac{π}{3}, \frac{5 π}{6}, \frac{4 π}{3}, \frac{11 π}{6}$
Find the critical points of $f (x) = \frac{\cos x}{2 + \sin x}$ on the open interval $(0, 2 π)$ $x = \frac{7 π}{6}, \frac{11 π}{6}$

How to do these

Find the critical points of $y = x^{3} - 12 x + 5$

Differentiate: the derivative is 3x squared minus 12.
Set that equal to zero: 3x squared equals 12, so x squared equals 4.
Solve: x equals 2 and x equals negative 2.

The answer is $x = - 2, x = 2$ .

Where students go wrong

Giving the y value instead of the x value. A critical point is answered with x, because it names a place along the axis, not a height. Some books want the pair, and the derivative gives you the x first either way.

Also called critical numbers, critical values or stationary points.

What you can put on this worksheet

Critical points: Find the critical points of $y = 2 x^{2} - 8$ → $x = 0$
Increasing and decreasing: $y = x^{3} - 3 x^{2} - 45 x - 3$ . Where is $y$ decreasing? → $(- 3, 5)$
Where f' is zero or does not exist: Find the critical points of $y = {(x - 5)}^{\frac{2}{3}}$ → $x = 5$
Trig and exponential functions on an interval: Find the critical points of $f (x) = \tan x - 2 x$ on the open interval $(0, 2 π)$ → $x = \frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$

Questions about these worksheets

Yes. Every critical 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 critical 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 4: critical points, increasing and decreasing, where f' is zero or does not exist and trig and exponential functions on an interval. Tick as many as you want and set how many of each, or let it spread them evenly.

Critical points worksheet

Select your difficulty

Find critical points

Find critical points, including undefined y′y'y′

Find critical points on (0,2π)(0, 2\pi)(0,2π)

Find critical points

Find critical points, including undefined y′y'y′

Find critical points on (0,2π)(0, 2\pi)(0,2π)

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these critical points worksheets free?+

Do the critical points worksheets come with an answer key?+

Are these critical points worksheets printable?+

What grade are these critical points worksheets for?+

Can I get a different version of the same critical points worksheet?+

Is this a free alternative to Kuta Software for critical points?+

Can I choose how hard the critical points questions are?+

What kinds of question are on a critical points worksheet?+

What do students get wrong with critical points?+

Find critical points, including undefined $y'$

Find critical points on $(0, 2\pi)$

Find critical points, including undefined $y'$

Find critical points on $(0, 2\pi)$