Verify That the Function Satisfies the Three Hypotheses of Rolle's $c$

question 105

Short Answer

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.
$$f ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 6 x + 1 , [ 0,4 ]$$

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Definitions:

Constant Returns to Scale

A situation in which the proportionate increase in inputs leads to an equal proportionate increase in outputs, meaning costs and outputs scale equally as production expands.

Diseconomies of Scale

A situation where a business's cost per unit increases as it produces more items, due to inefficiencies that arise with scaling up production.

Plant Size

Refers to the physical capacity or output level of a manufacturing or production facility.

Output Levels

The quantity of goods or services produced by a company, sector, or economy in a given period of time.