Local and absolute extrema

Answer Key - Grade 12 Math Worksheet

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

1. 1.

   Bo finds the absolute maximum value of f(x) = x³ − 3x² on −1 ≤ x ≤ 4. What is it?

   16
2. 2.

   Esme finds the absolute maximum value of f(x) = x³ − 27x on −4 ≤ x ≤ 2. What is it?

   54
3. 3.

   Jun finds the absolute maximum value of f(x) = x³ − 6x² on −2 ≤ x ≤ 5. What is it?

   0
4. 4.

   Kira uses the first derivative test on f(x) = xe^(−2x). At what x does f have its local maximum?

   1/2
5. 5.

   Luz uses the first derivative test on f(x) = xe^(−3x). At what x does f have its local maximum?

   1/3
6. 6.

   Noah uses the first derivative test on f(x) = xe^(−5x). At what x does f have its local maximum?

   1/5
7. 7.

   f′(x) = (x + 1)(x − 2)²(x − 4). Pia makes a sign chart for f′. Which describes every critical point of f?

   • local max at x = 2; local min at x = −1 and 4
   • local max at x = −1; local min at x = 4; neither at x = 2
   • local max at x = 4; local min at x = −1; neither at x = 2
8. 8.

   f′(x) = (x − 1)²(x + 3). Ren makes a sign chart for f′. Which describes every critical point of f?

   • local min at x = −3; neither at x = 1
   • local max at x = −3; neither at x = 1
   • local max at x = −3; local min at x = 1
9. 9.

   f′(x) = x(x − 3)². Tala makes a sign chart for f′. Which describes every critical point of f?

   • local max at x = 0; local min at x = 3
   • local min at x = 0; neither at x = 3
   • local max at x = 0; neither at x = 3
10. 10.

    f′(x) = (x + 4)(x − 1)(x − 5)². Zee makes a sign chart for f′. Which describes every critical point of f?

    • local max at x = −4; local min at x = 1; neither at x = 5
    • local max at x = 1; local min at x = −4 and 5
    • local max at x = 1; local min at x = −4; neither at x = 5