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Grade 12 · Math · Calculus

12th grade Critical points and where a function rises or falls worksheet

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Critical points and where a function rises or falls

Grade 12 Math Worksheet

NameDate

Circle the best answer, or write your answer on the line.

  1. Ari differentiates f(x) = x³ − 12x and makes a sign chart. Where is f increasing?

    • −2 < x < 2
    • x > −2
    • x > 2
    • x < −2 or x > 2

  2. Bo differentiates f(x) = x³ − 3x² − 9x and makes a sign chart. Where is f decreasing?

    • −1 < x < 3
    • x < 1
    • x < −1 or x > 3
    • x > −1

  3. Esme differentiates f(x) = x³ + 3x² − 9x and makes a sign chart. Where is f increasing?

    • x < −3 or x > 1
    • x > −3
    • −3 < x < 1
    • x > 1

  4. Kira finds the critical point of f(x) = x²e^(-4x) other than x = 0. Where is it?

  5. Luz finds the critical point of f(x) = x³e^(-2x) other than x = 0. Where is it?

  6. Milo finds the critical point of f(x) = x²e^(3x) other than x = 0. Where is it?

  7. Pia finds the critical points of f(x) = x + 2 sin x for 0 < x < 2π. Which are they?

    • x = π/3 and x = 5π/3
    • x = 7π/6 and x = 11π/6
    • x = 2π/3 and x = 4π/3
    • x = 2π/3

  8. Ren finds the critical points of f(x) = √2 x − 2 cos x for 0 < x < 2π. Which are they?

    • x = 5π/4 and x = 7π/4
    • x = 3π/4 and x = 5π/4
    • x = 5π/4
    • x = π/4 and x = 3π/4

  9. Tala finds the critical points of f(x) = √3 x − 2 sin x for 0 < x < 2π. Which are they?

    • x = π/3 and x = 2π/3
    • x = 5π/6 and x = 7π/6
    • x = π/6
    • x = π/6 and x = 11π/6

  10. Zee finds the critical points of f(x) = x − 2 cos x for 0 < x < 2π. Which are they?

    • x = π/6 and x = 5π/6
    • x = 7π/6
    • x = 2π/3 and x = 4π/3
    • x = 7π/6 and x = 11π/6
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12th grade Critical points and where a function rises or falls worksheet – printable
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About this worksheet

Critical points and where a function rises or falls. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can find critical points where f′ = 0 or f′ does not exist.
  • I can make a sign chart of f′ and read where f increases or decreases.
  • I can handle squared factors, trig functions and an unknown coefficient.

Standard: none (beyond CCSS)

Words to know

critical point: An x in the domain with f′ = 0 or where f′ does not exist.

sign chart: A number line marking where f′ is + or −.

increasing: Rising left to right: f′ > 0.

Common mistakes

Set f″ = 0. Critical points come from f′ = 0 or where f′ does not exist.

Found one value but missed another solution.

Swapped increasing and decreasing. f rises where f′ > 0 and falls where f′ < 0.

Questions parents ask

What is the most common mistake with critical points and where a function rises or falls?

Set f″ = 0. Critical points come from f′ = 0 or where f′ does not exist.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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