Critical points and where a function rises or falls

Answer Key - Grade 12 Math Worksheet - Challenge Me

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

1. 1.

   Ari finds every critical point of f(x) = x^(2/3)(x − 5), including any where f′ does not exist. List them, separated by commas.

   0, 2
2. 2.

   Bo finds every critical point of f(x) = x^(1/3)(x + 4), including any where f′ does not exist. List them, separated by commas.

   -1, 0
3. 3.

   Esme finds every critical point of f(x) = x^(2/3)(x − 10), including any where f′ does not exist. List them, separated by commas.

   0, 4
4. 4.

   f(x) = x³ + kx² + 5x. Luz wants a critical point at x = 1. What number k makes that happen?

   -4
5. 5.

   f(x) = x³ + kx² + 24x. Milo wants a critical point at x = 2. What number k makes that happen?

   -9
6. 6.

   f(x) = x³ + kx² + 8x. Noah wants a critical point at x = -2. What number k makes that happen?

   5
7. 7.

   f′(x) = (x − 1)²(x − 4). Pia makes a sign chart, careful with the squared factor. Where is f increasing?

   • x > 4
   • x > 1
   • x < 1 or x > 4
   • x < 4
8. 8.

   f′(x) = (x + 2)(x − 3)². Ren makes a sign chart, careful with the squared factor. Where is f increasing?

   • −2 < x < 3
   • x < −2
   • x > −2
   • x > 3
9. 9.

   f′(x) = x²(x + 5). Tala makes a sign chart, careful with the squared factor. Where is f increasing?

   • −5 < x < 0
   • x > 0
   • x < −5
   • x > −5
10. 10.

    f′(x) = (x − 2)²(x − 6). Zee makes a sign chart, careful with the squared factor. Where is f increasing?

    • x < 6
    • x > 2
    • x < 2 or x > 6
    • x > 6