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

question 20

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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Definitions:

Configuration

The fixed arrangement of atoms in a molecule that cannot be changed without breaking chemical bonds, different from conformation.

Asymmetric Carbon

A carbon atom bonded to four different groups or atoms, leading to chirality and optical activity in molecules.

Optical Activity

The ability of a substance to rotate the plane of polarization of polarized light, a property of chiral molecules.

Specific Rotation

A measure of the degree to which a chiral compound can rotate plane-polarized light, specific to the compound's concentration and temperature.