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

question 244

Multiple Choice

Use the function and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.Use a graphing utility to verify your results. $f ( x ) = 5 x ^ { 6 } + 6 x ^ { 5 } , \quad f ^ { \prime } ( x ) = 30 x ^ { 5 } + 30 x ^ { 4 }$

• A) \(f\) has vertical tangents at \(( - 1 , - 1 )\) and \(( 0,0 )\) .
• B) \(f\) has horizontal tangents at \(( 1,1 )\) and \(( 0,0 )\) .
• C) \(f\) has horizontal tangents at \(( - 1 , - 1 )\) and \(( 0,0 )\) .
• D) \(f\) has vertical tangents at \(( 1,1 )\) and \(( 0,0 )\) .
• E) \(f\) has vertical tangents at \(( - 1 , - 1 )\) and \(( 1,1 )\) .

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Definitions:

Fertility Rates

The average number of children a woman is expected to have during her lifetime, a key demographic measure affecting population growth.

Death Rates

The number of deaths in a particular population, scaled to the size of that population, per unit of time.

New Malthusians

A group or perspective that revisits Thomas Malthus's ideas concerning population growth, advocating for population control to prevent or mitigate future environmental and resource crises.

Reproductive Lives

The aspects of an individual or group's life that relate to reproduction, including all matters concerning fertility, childbirth, and family planning.