Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
### Determining Intervals of Concavity and Points of Inflection To determine the intervals on which the graph of \( y = f(x) \) is concave up or concave down, and to find the \( x \)-values at which the points of inflection occur, we follow the steps below using the function: \[ f(x) = x(x - 7\sqrt{x}), \quad x > 0 \] ### Points of Inflection (Enter an exact answer. Use symbolic notation and fractions where needed. Give your answer in the form of a comma-separated list if necessary. Enter DNE if there are no points of inflection.) \[ x = \] ### Intervals of Concavity #### Concave Up \[ f \text{ is concave up when } x \in \] #### Concave Down \[ f \text{ is concave down when } x \in \] ### Instructions for Notation - Use symbolic notation and fractions where needed. - Give your answers as intervals in the form \( (\ast, \ast) \). - Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and the appropriate type of parenthesis \( "(", ")", "[", \text{or} "]" \), depending on whether the interval is open or closed. - Enter \( \emptyset \) if the interval is empty. By following these steps, students will be able to clearly identify the intervals of concavity and the points of inflection for the given function.
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Transcribed Image Text:### Determining Intervals of Concavity and Points of Inflection To determine the intervals on which the graph of \( y = f(x) \) is concave up or concave down, and to find the \( x \)-values at which the points of inflection occur, we follow the steps below using the function: \[ f(x) = x(x - 7\sqrt{x}), \quad x > 0 \] ### Points of Inflection (Enter an exact answer. Use symbolic notation and fractions where needed. Give your answer in the form of a comma-separated list if necessary. Enter DNE if there are no points of inflection.) \[ x = \] ### Intervals of Concavity #### Concave Up \[ f \text{ is concave up when } x \in \] #### Concave Down \[ f \text{ is concave down when } x \in \] ### Instructions for Notation - Use symbolic notation and fractions where needed. - Give your answers as intervals in the form \( (\ast, \ast) \). - Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and the appropriate type of parenthesis \( "(", ")", "[", \text{or} "]" \), depending on whether the interval is open or closed. - Enter \( \emptyset \) if the interval is empty. By following these steps, students will be able to clearly identify the intervals of concavity and the points of inflection for the given function.
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