Critical points and where a function rises or falls

Answer Key - Grade 12 Math Worksheet

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

1. 1.

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

   • x < −3 or x > 1
   • x > −3
   • x > 1
   • −3 < x < 1
2. 2.

   Jun differentiates f(x) = 2x³ − 9x² + 12x and makes a sign chart. Where is f decreasing?

   • x > 1
   • x < 1 or x > 2
   • 1 < x < 2
   • x < 3/2
3. 3.

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

   1/2
4. 4.

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

   3/2
5. 5.

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

   -2/3
6. 6.

   Noah finds the critical point of f(x) = x⁴e^(-5x) other than x = 0. Where is it?

   4/5
7. 7.

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

   • x = 2π/3 and x = 4π/3
   • x = 7π/6 and x = 11π/6
   • x = π/3 and x = 5π/3
   • x = 2π/3
8. 8.

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

   • x = π/4 and x = 3π/4
   • x = 5π/4
   • x = 3π/4 and x = 5π/4
   • x = 5π/4 and x = 7π/4
9. 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 = π/6 and x = 11π/6
   • x = 5π/6 and x = 7π/6
   • x = π/6
10. 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 = 2π/3 and x = 4π/3
    • x = 7π/6 and x = 11π/6
    • x = 7π/6