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Worksheet: Concavity of a Function

Q1:

Use the given graph of 𝑓 to find all possible intervals on which 𝑓 is concave upward.

  • A 𝑓 is concave upward on ( 3 , 6 ) .
  • B 𝑓 is concave upward on ( 0 , 3 ) .
  • C 𝑓 is concave upward on ( 4 , 6 ) .
  • D 𝑓 is concave upward on ( 0 , 2 ) .
  • E 𝑓 is concave upward on ( 1 , 2 ) .

Q2:

Use the given graph of 𝑓 to find the coordinates of the points of inflection.

  • A ( 1 , 3 ) , ( 5 , 4 )
  • B ( 1 , 3 ) , ( 4 , 3 ) , ( 5 , 4 )
  • C ( 5 , 4 )
  • D ( 2 , 2 ) , ( 4 , 3 ) , ( 5 , 4 )
  • E ( 4 , 5 )

Q3:

Use the given graph of 𝑓 to find all possible intervals on which 𝑓 is concave downward.

  • A 𝑓 is concave downward on ( 0 , 2 ) .
  • B 𝑓 is concave downward on ( 2 , 4 ) and ( 5 , 7 ) .
  • C 𝑓 is concave downward on ( 4 , 5 ) .
  • D 𝑓 is concave downward on ( 0 , 2 ) and ( 4 , 5 ) .
  • E 𝑓 is concave downward on ( 0 , 1 ) .

Q4:

Use the given graph of 𝑓 ( π‘₯ ) to find the coordinates of the point of inflection.

  • A ( 3 , 6 )
  • B ( 1 , 2 )
  • C ( 4 , 1 )
  • D ( 2 , 3 )
  • E ( 3 , 4 ) , ( 4 , 1 )

Q5:

Use the given graph of 𝑓 to find all possible intervals on which 𝑓 is concave upward.

  • A 𝑓 is concave upward on ( 1 , 4 ) and ( 5 , 7 ) .
  • B 𝑓 is concave upward on ( 1 , 2 ) and ( 4 , 5 ) .
  • C 𝑓 is concave upward on ( 5 , 7 ) .
  • D 𝑓 is concave upward on ( 2 , 4 ) and ( 5 , 7 ) .
  • E 𝑓 is concave upward on ( 0 , 2 ) .

Q6:

Use the given graph of 𝑓 to find all possible intervals on which 𝑓 is concave downward.

  • A 𝑓 is concave downward on ( 3 , 6 ) .
  • B 𝑓 is concave downward on ( 0 , 2 ) .
  • C 𝑓 is concave downward on ( 4 , 6 ) .
  • D 𝑓 is concave downward on [ 2 , 4 ] and [ 4 , 6 ] .
  • E 𝑓 is concave downward on ( 1 , 6 ) .

Q7:

The graph of the first derivative 𝑓 β€² of a continuous function 𝑓 is shown. On what intervals is 𝑓 concave upward or concave downward?

  • A 𝑓 is concave upward on ( 1 , 6 ) and ( 8 , ∞ ) and concave downward on ( 0 , 1 ) and ( 6 , 8 ) .
  • B 𝑓 is concave upward on ( 2 , 3 ) and ( 5 , 7 ) and concave downward on ( 0 , 2 ) , ( 3 , 5 ) , and ( 7 , ∞ ) .
  • C 𝑓 is concave upward on ( 0 , 1 ) and ( 6 , 8 ) and concave downward on ( 1 , 6 ) and ( 8 , ∞ ) .
  • D 𝑓 is concave upward on ( 0 , 2 ) , ( 3 , 5 ) , and ( 7 , ∞ ) and concave downward on ( 2 , 3 ) and ( 5 , 7 ) .
  • E 𝑓 is concave upward on ( 2 , 4 ) and ( 6 , ∞ ) and concave downward on ( 0 , 2 ) and ( 4 , 6 ) .

Q8:

The graph of the first derivative 𝑓 β€² of a continuous function 𝑓 is shown. On what intervals is 𝑓 concave upward or concave downward?

  • A 𝑓 is concave upward on ( 1 , 5 ) and ( 6 , ∞ ) and concave downward on ( 0 , 1 ) and ( 5 , 6 ) .
  • B 𝑓 is concave upward on ( 0 , 3 ) and concave downward on ( 3 , 6 ) and ( 6 , ∞ ) .
  • C 𝑓 is concave upward on ( 0 , 1 ) and ( 5 , 6 ) and concave downward on ( 1 , 5 ) and ( 6 , ∞ ) .
  • D 𝑓 is concave upward on ( 3 , 6 ) and ( 6 , ∞ ) and concave downward on ( 0 , 3 ) .
  • E 𝑓 is concave upward on ( 0 , 2 ) , ( 4 , 6 ) , and ( 8 , ∞ ) and concave downward on ( 2 , 4 ) and ( 6 , 8 ) .