Use the Function Below and Its Derivative to Determine Any $f$

question 67

Multiple Choice

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.
$$f ( x ) = - 3 x ^ { 4 } + 6 x ^ { 2 } , f ^ { \prime } ( x ) = - 12 x ^ { 3 } + 12 x$$

• A) \(( - 1,3 ) , ( 0,0 ) , ( 1,3 )\)
• B) \(( - 1 , - 3 ) , ( 1 , - 12 )\)
• C) \(( - 1,3 ) , ( 1,3 )\)
• D) \(( 0,0 ) , ( 1,0 )\)
• E) \(( 0,0 ) , ( 1,3 )\)

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Definitions:

Variable Costs

Expenses that vary in relation to the amount of production or business activity.

Fixed Costs

Costs that do not change with the level of output produced, such as rent and salaries.

Break-Even Point

The financial analysis term where total costs and total revenues are equal, resulting in no net loss or gain for the business.

Contribution Margin

Contribution margin is the amount remaining from sales revenue after variable expenses are deducted, showing how much contributes to covering fixed costs and profit.