Critical points and where a function rises or falls

Answer Key - Grade 12 Math Worksheet - Building Confidence

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

1. 1.

   Esme finds every critical point of f(x) = x³ − 6x². List the x-values, separated by commas.

   0, 4
2. 2.

   Jun finds every critical point of f(x) = x³ + 6x² − 15x. List the x-values, separated by commas.

   -5, 1
3. 3.

   f′(x) = (x − 1)(x − 4). Kira makes a sign chart for f′. Where is f increasing?

   • x < 1 or x > 4
   • x > 1
   • 1 < x < 4
   • x > 4
4. 4.

   f′(x) = (x + 2)(x − 3). Luz makes a sign chart for f′. Where is f increasing?

   • −2 < x < 3
   • x < −2 or x > 3
   • x > 3
   • x > −2
5. 5.

   f′(x) = x(x − 5). Milo makes a sign chart for f′. Where is f increasing?

   • x > 0
   • 0 < x < 5
   • x > 5
   • x < 0 or x > 5
6. 6.

   f′(x) = (x + 4)(x + 1). Noah makes a sign chart for f′. Where is f increasing?

   • x < −4 or x > −1
   • x > −1
   • x > −4
   • −4 < x < −1
7. 7.

   Pia finds the critical point of f(x) = 4x + 9/x for x > 0. Where is it?

   3/2
8. 8.

   Ren finds the critical point of f(x) = 9x + 4/x for x > 0. Where is it?

   2/3
9. 9.

   Tala finds the critical point of f(x) = 3x + 48/x for x > 0. Where is it?

   4
10. 10.

    Zee finds the critical point of f(x) = 25x + 4/x for x > 0. Where is it?

    2/5