The Mean Value Theorem

Answer Key - Grade 12 Math Worksheet - Challenge Me

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

1. 1.

   f is differentiable, f(1) = 3, and f′(x) ≤ 5 for all x. Ari uses the Mean Value Theorem. What is the largest f(4) can be?

   18
2. 2.

   f is differentiable, f(0) = -1, and f′(x) ≤ 4 for all x. Bo uses the Mean Value Theorem. What is the largest f(6) can be?

   23
3. 3.

   f is differentiable, f(2) = 10, and f′(x) ≤ 3 for all x. Esme uses the Mean Value Theorem. What is the largest f(7) can be?

   25
4. 4.

   f is differentiable, f(-1) = 7, and f′(x) ≤ 2 for all x. Jun uses the Mean Value Theorem. What is the largest f(4) can be?

   17
5. 5.

   Kira finds the c from the Mean Value Theorem for f(x) = √x on 1 ≤ x ≤ 9. What is c?

   4
6. 6.

   Luz finds the c from the Mean Value Theorem for f(x) = √x on 4 ≤ x ≤ 16. What is c?

   9
7. 7.

   Noah finds the c from the Mean Value Theorem for f(x) = √x on 9 ≤ x ≤ 25. What is c?

   16
8. 8.

   f is differentiable on 1 ≤ x ≤ 5, with f(1) = 3 and f(5) = 11. Pia applies the Mean Value Theorem. Which must be true?

   • some c between 1 and 5 has f(c) = 2
   • f′(x) = 2 at each x between 1 and 5
   • some c between 1 and 5 has f′(c) = 8
   • some c between 1 and 5 has f′(c) = 2
9. 9.

   f is differentiable on 0 ≤ x ≤ 4, with f(0) = 10 and f(4) = -2. Ren applies the Mean Value Theorem. Which must be true?

   • some c between 0 and 4 has f(c) = -3
   • some c between 0 and 4 has f′(c) = -3
   • some c between 0 and 4 has f′(c) = -12
   • f′(x) = -3 at each x between 0 and 4
10. 10.

    f is differentiable on -2 ≤ x ≤ 4, with f(-2) = 5 and f(4) = 17. Tala applies the Mean Value Theorem. Which must be true?

    • f′(x) = 2 at each x between -2 and 4
    • some c between -2 and 4 has f′(c) = 2
    • some c between -2 and 4 has f′(c) = 12
    • some c between -2 and 4 has f(c) = 2