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Lesson: Concavity and Points of Inflection

In this lesson, we will learn how to determine the concavity of functions as well as their inflection points.

Worksheet • 25 Questions

Q1:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ + π‘₯ 5 3 is concave up and down.

  • A The function is concave up on ο€Ώ βˆ’ ∞ , βˆ’ √ 3 0 2 0  and ο€Ώ 0 , √ 3 0 2 0  and concave down on ο€Ώ βˆ’ √ 3 0 2 0 , 0  and ο€Ώ √ 3 0 2 0 , ∞  .
  • B The function is concave up on ο€Ώ βˆ’ √ 3 0 2 0 , 0  and ο€Ώ 0 , √ 3 0 2 0  and concave down on ο€Ώ βˆ’ ∞ , βˆ’ √ 3 0 2 0  and ο€Ώ √ 3 0 2 0 , ∞  .
  • C The function is concave up on ο€Ώ βˆ’ √ 3 0 2 0 , 0  and ο€Ώ √ 3 0 2 0 , ∞  and concave down on ο€Ώ βˆ’ ∞ , βˆ’ √ 3 0 2 0  and ο€Ώ 0 , √ 3 0 2 0  .
  • D The function is concave up on ο€Ώ 0 , √ 3 0 2 0  and ο€Ώ √ 3 0 2 0 , ∞  and concave down on ο€Ώ βˆ’ ∞ , βˆ’ √ 3 0 2 0  and ο€Ώ βˆ’ √ 3 0 2 0 , 0  .
  • E The function is concave up on ο€Ώ βˆ’ ∞ , βˆ’ √ 3 0 2 0  and ο€Ώ βˆ’ √ 3 0 2 0 , 0  and concave down on ο€Ώ 0 , √ 3 0 2 0  and ο€Ώ √ 3 0 2 0 , ∞  .

Q2:

Determine the intervals on which 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ + ( π‘₯ + 3 ) βˆ’ 4 5 is concave up and down.

  • A The function is concave down on the interval ( βˆ’ ∞ , βˆ’ 3 ) and up on the interval ( βˆ’ 3 , ∞ ) .
  • B The function is concave down on the interval ( βˆ’ ∞ , 3 ) and up on the interval ( 3 , ∞ ) .
  • C The function is concave down on the interval ( βˆ’ 3 , ∞ ) and up on the interval ( βˆ’ ∞ , βˆ’ 3 ) .
  • D The function is concave down on the interval ( βˆ’ 4 , ∞ ) and up on the interval ( βˆ’ ∞ , 1 ) .
  • E The function is concave down on the interval ( βˆ’ ∞ , 1 ) and up on the interval ( βˆ’ 4 , ∞ ) .

Q3:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = βˆ’ 3 π‘₯ + √ 9 π‘₯ + 1 2 is concave up and down.

  • A The function is concave up on ( βˆ’ ∞ , ∞ ) .
  • BThe function is concave down on ( βˆ’ 9 , ∞ ) .
  • CThe function is concave down on ( βˆ’ ∞ , ∞ ) .
  • DThe function is concave down on ( 0 , ∞ ) .
  • EThe function is concave up on ( βˆ’ 9 , ∞ ) .
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