Verify That the Function Satisfies the Three Hypotheses of Rolle's $$f ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 6 x + 1 , [ 0,4 ]$$

question 56

Multiple Choice

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.
$$f ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 6 x + 1 , [ 0,4 ]$$

• A)
• \[c = 2 + \frac { 2 \sqrt { 3 } } { 3 }\]
• B)
• \[c _ { 1 } = \frac { 5 } { 3 } + \frac { \sqrt { 7 } } { 3 } , c _ { 2 } = \frac { 5 } { 3 } - \frac { \sqrt { 7 } } { 3 }\]
• C)
• \[c _ { 1 } = 12 + \frac { \sqrt { 2 } } { 3 } , c _ { 2 } = 12 - \frac { \sqrt { 2 } } { 3 }\]
• D)
• \[c = 2 - \frac { 2 \sqrt { 3 } } { 3 }\]
• E)
• \[c _ { 1 } = 2 + \frac { 2 \sqrt { 3 } } { 3 } , c _ { 2 } = \frac { 2 \sqrt { 3 } } { 3 }\]

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