Determine Whether Rolle's Theorem Can Be Applied to the Function $f ( x ) = ( x - 5 ) ( x - 6 ) ( x - 7 )$

question 16

Multiple Choice

Determine whether Rolle's Theorem can be applied to the function $f ( x ) = ( x - 5 ) ( x - 6 ) ( x - 7 )$ on the closed interval $[ 5,7 ]$ . If Rolle's Theorem can be applied, find all numbers $c$ in the open interval $( 5,7 )$ such that $f ^ { t } ( c ) = 0$ .

• A)Rolle's Theorem applies; \(c = 5 + \frac { \sqrt { 3 } } { 3 }\)
• B)Rolle's Theorem applies; \(c = 6 + \frac { \sqrt { 3 } } { 3 }\)
• C)Rolle's Theorem does apply; \(c = 7 - \frac { \sqrt { 3 } } { 3 }\)
• D)Rolle's Theorem applies; \(c = 6 + \frac { \sqrt { 3 } } { 3 } , 6 - \frac { \sqrt { 3 } } { 3 }\)
• E) Rolle's Theorem does not apply

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Definitions:

Short Run

A period in economic analysis during which at least one input, such as plant size or capital, is fixed, limiting the ability to adjust to demand changes.

Long Run

A period in which all factors of production and costs are variable, allowing for full adjustment to changes in the economic environment.

Specialization

The process of focusing effort and resources on a limited number of activities, goods, or services to gain a comparative advantage in production.

Decreasing Returns To Scale

Long-run average total cost increases as output increases (also known as diseconomies of scale).