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Worksheet: Concavity and Points of Inflection

In this worksheet, we will practice determining the concavity of functions as well as their inflection points.

Q1:

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

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

Q2:

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

  • AThe function is concave down on ο€Ώ βˆ’ ∞ , βˆ’ √ 6 6  and ο€Ώ βˆ’ √ 6 6 , √ 6 6  and concave up on ο€Ώ √ 6 6 , ∞  .
  • BThe function is concave down on ο€Ώ βˆ’ √ 6 6 , √ 6 6  and concave up on ο€Ώ βˆ’ ∞ , βˆ’ √ 6 6  and ο€Ώ √ 6 6 , ∞  .
  • CThe function is concave down on ο€Ώ √ 6 6 , ∞  and concave up on ο€Ώ βˆ’ ∞ , βˆ’ √ 6 6  and ο€Ώ βˆ’ √ 6 6 , √ 6 6  .
  • DThe function is concave down on ο€Ώ βˆ’ ∞ , βˆ’ √ 6 6  and ο€Ώ √ 6 6 , ∞  and concave up on ο€Ώ βˆ’ √ 6 6 , √ 6 6  .
  • EThe function is concave down on ο€Ώ βˆ’ √ 6 6 , √ 6 6  and ο€Ώ √ 6 6 , ∞  and concave up on ο€Ώ βˆ’ ∞ , βˆ’ √ 6 6  .

Q3:

Find the inflection points of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 4 π‘₯ + 1 2 2 .

  • A The inflection point is ο€Ό 1 4 , βˆ’ 3 4  .
  • B The inflection points are ο€Ό 1 2 , βˆ’ 3 8  and ο€Ό βˆ’ 1 2 , βˆ’ 3 8  .
  • C The inflection point is ο€Ό βˆ’ 4 , 3 1 3  .
  • D The inflection points are ο€Ώ √ 3 6 , βˆ’ 1 1 1 6  and ο€Ώ βˆ’ √ 3 6 , βˆ’ 1 1 1 6  .
  • E The inflection points are ο€Ώ √ 3 2 , βˆ’ 1 1 6  and ο€Ώ βˆ’ √ 3 2 , βˆ’ 1 1 6  .

Q4:

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

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

Q5:

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

Q6:

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

Q7:

Find the intervals over which the graph of the function is convex downwards and convex upwards.

  • Aconvex upwards over the interval , convex downwards over the interval
  • Bconvex downwards over the interval , convex upwards over the interval
  • Cconvex upwards over the intervals and , convex downwards over the interval
  • Dconvex downwards over the intervals and , convex upwards over the interval

Q8:

Find the inflection point of the function 𝑓 ( π‘₯ ) = βˆ’ 5 π‘₯ + ( π‘₯ βˆ’ 4 ) + 2 5 .

  • A The inflection point is ( 4 , βˆ’ 2 0 ) .
  • B The inflection point is ( 4 , 1 8 ) .
  • C The inflection point is ( 3 , βˆ’ 1 4 ) .
  • D The inflection point is ( 4 , βˆ’ 1 8 ) .
  • E The inflection point is ( 5 , βˆ’ 2 2 ) .

Q9:

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

Q10:

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 βˆ’ √ 3 0 2 0 , 0  and concave down on ο€Ώ 0 , √ 3 0 2 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 √ 3 0 2 0 , ∞  and concave down on ο€Ώ βˆ’ ∞ , βˆ’ √ 3 0 2 0  and ο€Ώ 0 , √ 3 0 2 0  .
  • C 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  .
  • D 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 , ∞  .
  • E 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 , ∞  .

Q11:

Given that 𝑓 ( π‘₯ ) = 4 π‘₯ + 4 π‘₯ s i n c o s , where 0 ≀ π‘₯ ≀ πœ‹ 2 , determine the inflection points of 𝑓 .

  • A 𝑓 has inflection points at ο€Ό 3 πœ‹ 1 6 , √ 2  and ο€Ό 7 πœ‹ 1 6 , √ 2  .
  • B 𝑓 has inflection points at ο€Ό 3 πœ‹ 1 6 , √ 2  and ο€Ό 7 πœ‹ 1 6 , βˆ’ √ 2  .
  • C 𝑓 has inflection points at ο€» πœ‹ 1 6 , √ 2  and ο€Ό 5 πœ‹ 1 6 , βˆ’ √ 2  .
  • D 𝑓 has inflection points at ο€Ό 3 πœ‹ 1 6 , 0  and ο€Ό 7 πœ‹ 1 6 , 0  .
  • E 𝑓 has inflection points at ο€» πœ‹ 1 6 , 0  and ο€Ό 5 πœ‹ 1 6 , 0  .

Q12:

For 0 ≀ π‘₯ ≀ 4 πœ‹ , find all the inflection points of 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ π‘₯ s i n .

  • A ( πœ‹ , 2 πœ‹ ) , ( 2 πœ‹ , 2 πœ‹ ) , ( 3 πœ‹ , 6 πœ‹ )
  • B ( πœ‹ , πœ‹ ) , ( 2 πœ‹ , 2 πœ‹ ) , ( 3 πœ‹ , 3 πœ‹ )
  • C ( πœ‹ , 2 πœ‹ ) , ( 2 πœ‹ , 4 πœ‹ ) , ( 3 πœ‹ , 3 πœ‹ )
  • D ( πœ‹ , 2 πœ‹ ) , ( 2 πœ‹ , 4 πœ‹ ) , ( 3 πœ‹ , 6 πœ‹ )
  • E ( πœ‹ , πœ‹ ) , ( 2 πœ‹ , 4 πœ‹ ) , ( 3 πœ‹ , 6 πœ‹ )

Q13:

Find all the inflection points of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 2 π‘₯ + 5 4 2 .

  • Ainflection points at ο€Ώ √ 3 3 , 4 0 9  and ο€Ώ βˆ’ √ 3 3 , βˆ’ 4 0 9 
  • Binflection points at ο€Ώ √ 3 3 , βˆ’ 4 0 9  and ο€Ώ βˆ’ √ 3 3 , βˆ’ 4 0 9 
  • Cinflection points at ( 1 , 4 ) , ( βˆ’ 1 , 4 ) and ( 0 , 5 )
  • Dinflection points at ο€Ώ √ 3 3 , 4 0 9  and ο€Ώ βˆ’ √ 3 3 , 4 0 9 
  • Einflection points at ( 1 , βˆ’ 4 ) , ( βˆ’ 1 , βˆ’ 4 ) and ( 0 , 5 )

Q14:

Find the inflection point on the curve 𝑦 = 6 π‘₯ ( π‘₯ + 1 ) 2 .

  • A ο€Ό βˆ’ 1 3 , βˆ’ 8 9 
  • B ο€Ό 2 3 , 1 0 0 9 
  • C ο€Ό βˆ’ 3 2 , βˆ’ 9 4 
  • D ο€Ό βˆ’ 2 3 , βˆ’ 4 9 
  • Ehas no inflection points

Q15:

Find the inflection point on the curve 𝑦 = 9 π‘₯ ( π‘₯ βˆ’ 1 ) 2 .

  • A ο€Ό 1 3 , 4 3 
  • B ο€Ό βˆ’ 2 3 , βˆ’ 5 0 3 
  • C ο€Ό 3 2 , 2 7 8 
  • D ο€Ό 2 3 , 2 3 
  • Ehas no inflection points

Q16:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = 3 π‘₯ 2 π‘₯ 2 l n is concave upward and downward.

  • AThe function is concave upward on ο€Ί 2 √ 𝑒 , ∞  and concave downward on ο€Ί 0 , 2 √ 𝑒  .
  • BThe function is concave upward on  0 , 𝑒 2  βˆ’ 3 2 and concave downward on  𝑒 2 , ∞  βˆ’ 3 2 .
  • CThe function is concave upward on ο€Ώ 1 2 √ 𝑒 , ∞  and concave downward on ο€Ώ 0 , 1 2 √ 𝑒  .
  • DThe function is concave upward on  𝑒 2 , ∞  βˆ’ 3 2 and concave downward on  0 , 𝑒 2  βˆ’ 3 2 .
  • EThe function is concave downward on ο€Ώ 0 , 1 2 √ 𝑒  and concave downward on ο€Ώ 1 2 √ 𝑒 , ∞  .

Q17:

A good definition of the function 𝑓 being concave up on an interval 𝐽 = ( π‘Ž , 𝑏 ) is that 𝑓 β€² is increasing on the interval. So the slope of the graph gets larger as π‘₯ increases.

If 𝑓 β€² β€² exists on the interval, what result would prove that 𝑓 is concave up if 𝑓 β€² β€² ( π‘₯ ) > 0 for π‘₯ in 𝐽 ?

  • Athe fact that if a function is negative at one point and positive at another, then it must be zero in between those points
  • Bthe fact that if the derivative of a function is zero, then the function attains a local maximum or minimum there
  • Cthe fact that a function has the same instantaneous rate of change at some point as its average rate of change over the interval
  • Dthe fact that if the derivative of a function is positive on an interval, then the function is increasing there

Consider the function 𝑔 ( π‘₯ ) = π‘₯ 4 . Is 𝑔 β€² increasing on the interval ( βˆ’ 1 , 1 ) ?

  • Ayes
  • Bno

With the function above, our definition says that the function 𝑔 is concave up on ( βˆ’ 1 , 1 ) . Is 𝑔 β€² β€² ( π‘₯ ) > 0 on this interval?

  • Ayes
  • Bno

Is it true that if 𝑓 is concave up on an interval, then 𝑓 β€² β€² ( π‘₯ ) > 0 on the interval? (Recall the definition above!)

  • Ayes
  • Bno

Q18:

Find (if any) the inflection points of 𝑓 ( π‘₯ ) = 3 π‘₯ 2 π‘₯ 2 l n .

  • A 𝑓 has an inflection point at ο€Ώ 1 2 √ 𝑒 , βˆ’ 3 8 𝑒  .
  • B 𝑓 has an inflection point at  𝑒 2 , 9 8 𝑒  βˆ’ 3 3 2 .
  • C 𝑓 has an inflection point at ο€Ώ 1 2 √ 𝑒 , 3 8 𝑒  .
  • D 𝑓 has an inflection point at  𝑒 2 , βˆ’ 9 8 𝑒  βˆ’ 3 3 2 .
  • E 𝑓 has no inflection points.

Q19:

Find the inflection point on the graph of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 9 π‘₯ + 6 π‘₯ 3 2 .

  • A ( 3 , 0 )
  • Bno inflection point
  • C ( 3 , βˆ’ 2 1 )
  • D ( 3 , βˆ’ 3 6 )

Q20:

Find the intervals over which the graph of the function is convex downwards and convex upwards.

  • A The graph is convex upwards over the interval , and is convex upwards over the interval .
  • B The graph is convex downwards over the interval , and is convex upwards over the interval .
  • C The graph is convex downwards over the interval , and is convex downwards over the interval .
  • D The graph is convex upwards over the interval , and is convex downwards over the interval .

Q21:

Find the inflection points of the function 𝑓 ( π‘₯ ) = π‘₯ 2 βˆ’ 3 π‘₯ + 4 4 2 .

  • A The inflection point are ο€Ό 1 , βˆ’ 5 2  and ο€Ό βˆ’ 1 , βˆ’ 5 2  .
  • B The inflection point are ( 1 , βˆ’ 4 ) and ( βˆ’ 1 , 4 ) .
  • C The inflection point are ( 1 , βˆ’ 2 ) and ( βˆ’ 1 , 2 ) .
  • D The inflection points are ο€Ό 1 , 3 2  and ο€Ό βˆ’ 1 , 3 2  .
  • E The inflection point are ο€Ό 1 , 1 1 2  and ο€Ό βˆ’ 1 , 1 1 2  .

Q22:

Find the inflection point of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 2 π‘₯ βˆ’ 1 3 .

  • AThe inflection point is ο€» 2 √ 3 , βˆ’ 1  .
  • BThe inflection point is ( 2 , βˆ’ 1 7 ) .
  • CThe inflection point is ( 0 , βˆ’ 1 2 ) .
  • DThe inflection point is ( 0 , βˆ’ 1 ) .
  • EThe inflection point is ( 0 , 2 ) .

Q23:

Find the inflection points of 𝑓 ( π‘₯ ) = βˆ’ 2 π‘₯ + 5 π‘₯ 5 3 .

  • A ο€Ώ √ 3 2 , 4 5 8  , ο€Ώ βˆ’ √ 3 2 , βˆ’ 4 5 8  , ( 0 , 3 ) .
  • B ο€Ώ √ 3 2 , βˆ’ 3 √ 3 3 2  , ο€Ώ βˆ’ √ 3 2 , 3 √ 3 3 2  .
  • C ο€Ώ √ 3 2 , 4 5 8  , ο€Ώ βˆ’ √ 3 2 , βˆ’ 4 5 8  .
  • D ο€Ώ √ 3 2 , 2 1 √ 3 1 6  , ο€Ώ βˆ’ √ 3 2 , βˆ’ 2 1 √ 3 1 6  , ( 0 , 0 ) .
  • E ο€Ώ 5 √ 6 6 , 4 7 5 3 6  , ο€Ώ βˆ’ 5 √ 6 6 , βˆ’ 4 7 5 3 6  , ( 0 , βˆ’ 3 ) .

Q24:

Find, if any, the inflection points on the graph of

  • A ( 0 , βˆ’ 7 )
  • B T h e f u n c t i o n h a s n o i n fl e c t i o n p o i n t s .
  • C ( 0 , 0 )
  • D ( 0 , 4 )

Q25:

The curve 𝑦 = π‘˜ π‘₯ + π‘₯ βˆ’ 5 2 3 has an inflection point at π‘₯ = 1 . What is π‘˜ ?

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