Determine Where the Graph of the Function $f ( x ) = x ^ { 3 } - 6 x$

question 37

Multiple Choice

Determine where the graph of the function $f ( x ) = x ^ { 3 } - 6 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

• A) \(\mathrm { CU }\) on \(( \sqrt { 2 } , \infty ) , \mathrm { CD }\) on \(( - \infty , \sqrt { 2 } )\),
• IP \(( \sqrt { 2 } , - 4 \sqrt { 2 } )\)
• B) \(\mathrm { CU }\) on \(( - \sqrt { 2 } , \sqrt { 2 } ) , \mathrm { CD }\) on \(( - \infty , - \sqrt { 2 } )\) and \(( \sqrt { 2 } , \infty )\)
• IP \(( - \sqrt { 2 } , 4 \sqrt { 2 } )\) and \(( \sqrt { 2 } , - 4 \sqrt { 2 } )\)
• C) \(\mathrm { CU }\) on \(( 0 , \infty ) , \mathrm { CD }\) on \(( - \infty , 0 )\),
• IP \(( 0,0 )\)
• D) \(\operatorname { CU }\) on \(( - \sqrt { 2 } , \infty ) , \mathrm { CD }\) on \(( - \infty , - \sqrt { 2 } )\),
• IP \(( - \sqrt { 2 } , 4 \sqrt { 2 } )\)

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Definitions:

Direct Proportion

A relationship between two variables where an increase in one results in a proportional increase in the other.

Activity Level

In accounting and management, refers to the volume of units produced, hours worked, or any other measure of output that drives the costs in a business.

Production Supervisor's Salary

An overhead expense associated with the compensation paid to an individual responsible for overseeing and managing the production operations.

Units Produced

Refers to the total quantity of goods manufactured by a business during a specific period.