The Mean Value Theorem and Rolle's theorem practice problems
Check the hypotheses, then find every c where the tangent slope equals the average slope. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Verify that \( \displaystyle f(x) = x^{3} - 5 x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-1, 3] \), and find every number \( \displaystyle c \) the theorem guarantees.
Verify that \( \displaystyle f(x) = x^{3} - x^{2} - 2 x + 2 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [0, 3] \), and find every number \( \displaystyle c \) the theorem guarantees.
Verify that \( \displaystyle f(x) = 2 x^{3} + 4 x^{2} - 22 x - 24 \) satisfies the hypotheses of Rolle's theorem on \( \displaystyle [-1, 3] \), and find every number \( \displaystyle c \) the theorem guarantees.
Verify that \( \displaystyle f(x) = - x^{3} - 2 x^{2} - x - 2 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-1, 1] \), and find every number \( \displaystyle c \) the theorem guarantees.
Verify that \( \displaystyle f(x) = 2 x^{3} - 10 x^{2} + 12 x \) satisfies the hypotheses of Rolle's theorem on \( \displaystyle [0, 2] \), and find every number \( \displaystyle c \) the theorem guarantees.
Verify that \( \displaystyle f(x) = x^{3} - 5 x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [0, 3] \), and find every number \( \displaystyle c \) the theorem guarantees.
Verify that \( \displaystyle f(x) = - 2 x^{3} + x^{2} + x + 1 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [0, 4] \), and find every number \( \displaystyle c \) the theorem guarantees.
Verify that \( \displaystyle f(x) = - 2 x^{2} + 3 x - 4 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-1, 3] \), and find every number \( \displaystyle c \) the theorem guarantees.
Verify that \( \displaystyle f(x) = - x^{3} + x^{2} - x - 1 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [0, 3] \), and find every number \( \displaystyle c \) the theorem guarantees.
Verify that \( \displaystyle f(x) = x^{3} - 3 x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-2, 1] \), and find every number \( \displaystyle c \) the theorem guarantees.