Find the Points of Inflection and Discuss the Concavity of the Function

question 86

Multiple Choice

Find the points of inflection and discuss the concavity of the function $f ( x ) = - 8 x - 2 \cos x$ on the interval $[ 0,2 \pi ]$

• A) concave down on \(( 0,2 \pi )\); no points of inflection
• B) concave downward on \(\left( \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right) ;\) concave upward on \(\left( 0 , \frac { \pi } { 2 } \right) , \left( \frac { 3 \pi } { 2 } , 2 \pi \right)\); inflection points at \(x = \frac { \pi } { 2 }\) and \(x = \frac { 3 \pi } { 2 }\)
• C) concave upward on \(\left( \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right) ;\) concave downward on \(\left( 0 , \frac { \pi } { 2 } \right) , \left( \frac { 3 \pi } { 2 } , 2 \pi \right)\) inflection points at \(x = \frac { \pi } { 2 }\) and \(x = \frac { 3 \pi } { 2 }\)
• D) concave up on \(( 0,2 \pi )\); no points of inflection
• E) none of the above

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Definitions:

Verstehen

A German term used by Max Weber that refers to the understanding of social behavior through empathetic or interpretive means, aiming to grasp the subjective meanings and motivations of social actors.

Organic Solidarity

A concept in sociology where social cohesion comes from the interdependence of individuals in more advanced societies, as described by Émile Durkheim.

Durkheim

A foundational figure in sociology who studied the social basis of human behavior and the role of social institutions in maintaining societal cohesion.

Mechanical Solidarity

Describes early societies based on similarities and independence.