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Lesson: Concavity and Points of Inflection

Worksheet • 25 Questions

Q1:

Find (if any) the inflection points of 𝑓 ( π‘₯ ) = 𝑒 βˆ’ 2 𝑒 + 5 π‘₯ π‘₯ .

  • AThere are no inflection points.
  • BThe inflection point is ο€Ώ 1 5 , √ 𝑒 2 √ 𝑒 + 5  5 5 .
  • CThe inflection point is ο€Ό 0 , 1 3  .
  • DThe inflection point is ο€Ώ 1 5 , √ 𝑒 βˆ’ 2 √ 𝑒 + 5  5 5 .
  • EThe inflection point is ο€Ό 0 , 1 7  .

Q2:

Determine where 𝑓 ( π‘₯ ) = π‘₯ 2 βˆ’ 3 π‘₯ + 3 4 2 is concave up and where it is concave down.

  • A The function is concave up on the intervals ( βˆ’ ∞ , βˆ’ 1 ) and ( 1 , ∞ ) and down on the interval ( βˆ’ 1 , 1 ) .
  • B The function is concave up on the intervals ( βˆ’ 1 , 1 ) and ( 1 , ∞ ) and down on the interval ( βˆ’ ∞ , βˆ’ 1 ) .
  • C The function is concave up on the interval ( βˆ’ 1 , 1 ) and down on the intervals ( βˆ’ ∞ , βˆ’ 1 ) and ( 1 , ∞ ) .
  • D The function is concave up on the intervals ( βˆ’ ∞ , βˆ’ 1 ) and ( βˆ’ 1 , 1 ) and down on the interval ( 1 , ∞ ) .
  • E The function is concave up on the interval ( 1 , ∞ ) and down on the intervals ( βˆ’ ∞ , βˆ’ 1 ) and ( βˆ’ 1 , 1 ) .

Q3:

For 0 < π‘₯ < 2 πœ‹ , determine the intervals on which 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 2 π‘₯ c o s s i n 2 is concave up and concave down.

  • A 𝑓 is concave up on the interval ο€Ό πœ‹ 6 , 5 πœ‹ 6  and concave down on the intervals ο€» 0 , πœ‹ 6  and ο€Ό 5 πœ‹ 6 , 2 πœ‹  .
  • B 𝑓 is concave up on the interval ( πœ‹ , 2 πœ‹ ) and concave down on the interval ( 0 , πœ‹ ) .
  • C 𝑓 is concave up on the intervals ο€» 0 , πœ‹ 6  and ο€Ό 5 πœ‹ 6 , 2 πœ‹  and concave down on the interval ο€Ό πœ‹ 6 , 5 πœ‹ 6  .
  • D 𝑓 is concave up on the intervals ο€» 0 , πœ‹ 2  and ο€Ό 3 πœ‹ 2 , 2 πœ‹  and concave down on the interval ο€Ό πœ‹ 6 , 5 πœ‹ 6  .
  • E 𝑓 is concave up on the interval ο€Ό πœ‹ 6 , 5 πœ‹ 6  and concave down on the intervals ο€» 0 , πœ‹ 2  and ο€Ό 3 πœ‹ 2 , 2 πœ‹  .
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