Worksheet: Concavity and Points of Inflection

In this worksheet, we will practice determining the concavity of a function as well as its inflection points using its second derivative.

Q1:

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

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

Q2:

Determine the inflection points of the curve 𝑦=π‘₯+2π‘₯βˆ’5.

  • A ( 0 , 2 )
  • B ( 2 , 3 )
  • Chas no inflection points
  • D ( βˆ’ 1 , 1 )

Q3:

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

  • AThe function is concave up on ο€Ώβˆ’βˆš3020,0 and ο€Ώ0,√3020 and concave down on ο€Ώβˆ’βˆž,βˆ’βˆš3020 and ο€Ώβˆš3020,βˆžο‹ .
  • BThe function is concave up on ο€Ώβˆ’βˆž,βˆ’βˆš3020 and ο€Ώ0,√3020 and concave down on ο€Ώβˆ’βˆš3020,0 and ο€Ώβˆš3020,βˆžο‹.
  • CThe function is concave up on ο€Ώβˆ’βˆš3020,0 and ο€Ώβˆš3020,βˆžο‹ and concave down on ο€Ώβˆ’βˆž,βˆ’βˆš3020 and ο€Ώ0,√3020.
  • DThe function is concave up on ο€Ώ0,√3020 and ο€Ώβˆš3020,βˆžο‹ and concave down on ο€Ώβˆ’βˆž,βˆ’βˆš3020 and ο€Ώβˆ’βˆš3020,0.
  • EThe function is concave up on ο€Ώβˆ’βˆž,βˆ’βˆš3020 and ο€Ώβˆ’βˆš3020,0 and concave down on ο€Ώ0,√3020 and ο€Ώβˆš3020,βˆžο‹ .

Q4:

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

  • AThe function is concave down on the interval (βˆ’βˆž,1) and up on the interval (βˆ’4,∞).
  • BThe function is concave down on the interval (βˆ’βˆž,3) and up on the interval (3,∞).
  • CThe function is concave down on the interval (βˆ’βˆž,βˆ’3) and up on the interval (βˆ’3,∞).
  • DThe function is concave down on the interval (βˆ’3,∞) and up on the interval (βˆ’βˆž,βˆ’3).
  • EThe function is concave down on the interval (βˆ’4,∞) and up on the interval (βˆ’βˆž,1).

Q5:

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

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

Q6:

Determine the intervals on which the function 𝑓(π‘₯)=4π‘₯βˆ’33π‘₯+1 is concave up and down.

  • AThe function is concave down on ο€Όβˆ’βˆž,βˆ’13 and ο€Όβˆ’13,13 and concave up on ο€Ό13,∞.
  • BThe function is concave down on ο€Όβˆ’13,13 and ο€Ό13,∞ and concave up on ο€Όβˆ’βˆž,βˆ’13.
  • CThe function is concave down on ο€Όβˆ’13,13 and concave up on ο€Όβˆ’βˆž,βˆ’13 and ο€Ό13,∞.
  • DThe function is concave down on ο€Ό13,∞ and concave up on ο€Όβˆ’βˆž,βˆ’13 and ο€Όβˆ’13,13.
  • EThe function is concave down on ο€Όβˆ’βˆž,βˆ’13 and ο€Ό13,∞ and concave up on ο€Όβˆ’13,13.

Q7:

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

  • A 𝑓 is concave up on the interval ο€Όπœ‹6,5πœ‹6 and concave down on the intervals ο€»0,πœ‹6 and ο€Ό5πœ‹6,2πœ‹οˆ.
  • B 𝑓 is concave up on the interval ο€Όπœ‹6,5πœ‹6 and concave down on the intervals ο€»0,πœ‹2 and ο€Ό3πœ‹2,2πœ‹οˆ.
  • C 𝑓 is concave up on the intervals ο€»0,πœ‹2 and ο€Ό3πœ‹2,2πœ‹οˆ and concave down on the interval ο€Όπœ‹6,5πœ‹6.
  • D 𝑓 is concave up on the intervals ο€»0,πœ‹6 and ο€Ό5πœ‹6,2πœ‹οˆ and concave down on the interval ο€Όπœ‹6,5πœ‹6.
  • E 𝑓 is concave up on the interval (πœ‹,2πœ‹) and concave down on the interval (0,πœ‹).

Q8:

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

  • AThe function is concave upward on 𝑒2,∞ and concave downward on 0,𝑒2.
  • BThe function is concave upward on ο€Ώ0,12βˆšπ‘’ο‹ and concave downward on ο€Ώ12βˆšπ‘’,βˆžο‹.
  • CThe function is concave upward on ο€Ί2βˆšπ‘’,βˆžο† and concave downward on ο€Ί0,2βˆšπ‘’ο†.
  • DThe function is concave upward on ο€Ώ12βˆšπ‘’,βˆžο‹ and concave downward on ο€Ώ0,12βˆšπ‘’ο‹.
  • EThe function is concave upward on 0,𝑒2 and concave downward on 𝑒2,∞.

Q9:

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

  • A ( 3 , βˆ’ 3 6 )
  • B ( 3 , βˆ’ 2 1 )
  • C ( 3 , 0 )
  • Dno inflection point

Q10:

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

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

Q11:

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

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

Q12:

Given that 𝑓(π‘₯)=4π‘₯+4π‘₯sincos, where 0≀π‘₯β‰€πœ‹2, determine the inflection points of 𝑓.

  • A 𝑓 has inflection points at ο€Ό3πœ‹16,√2 and ο€Ό7πœ‹16,βˆ’βˆš2.
  • B 𝑓 has inflection points at ο€Ό3πœ‹16,√2 and ο€Ό7πœ‹16,√2.
  • C 𝑓 has inflection points at ο€Ό3πœ‹16,0 and ο€Ό7πœ‹16,0.
  • D 𝑓 has inflection points at ο€»πœ‹16,√2 and ο€Ό5πœ‹16,βˆ’βˆš2.
  • E 𝑓 has inflection points at ο€»πœ‹16,0 and ο€Ό5πœ‹16,0.

Q13:

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

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

Q14:

Find the inflection points of 𝑓(π‘₯)=π‘₯βˆ’14π‘₯+1.

  • AThe inflection points are ο€Ώβˆš32,βˆ’116 and ο€Ώβˆ’βˆš32,βˆ’116.
  • BThe inflection point is ο€Ό14,βˆ’34.
  • CThe inflection point is ο€Όβˆ’4,313.
  • DThe inflection points are ο€Ώβˆš36,βˆ’1116 and ο€Ώβˆ’βˆš36,βˆ’1116.
  • EThe inflection points are ο€Ό12,βˆ’38 and ο€Όβˆ’12,βˆ’38.

Q15:

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

  • AThe inflection point is ο€Ό0,17.
  • BThe inflection point is ο€Ό0,13.
  • CThere are no inflection points.
  • DThe inflection point is ο€Ώ15,βˆšπ‘’2βˆšπ‘’+5ο‹οŽ€οŽ€.
  • EThe inflection point is ο€Ώ15,βˆšπ‘’βˆ’2βˆšπ‘’+5ο‹οŽ€οŽ€.

Q16:

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

  • A 𝑓 has an inflection point at 𝑒2,98π‘’οοŠ±οŠ©οŽ’οŽ‘.
  • B 𝑓 has no inflection points.
  • C 𝑓 has an inflection point at 𝑒2,βˆ’98π‘’οοŠ±οŠ©οŽ’οŽ‘.
  • D 𝑓 has an inflection point at ο€Ώ12βˆšπ‘’,βˆ’38𝑒.
  • E 𝑓 has an inflection point at ο€Ώ12βˆšπ‘’,38𝑒.

Q17:

Find, if any, the inflection points on the graph of 𝑓(π‘₯)=6π‘₯βˆ’7π‘₯+4π‘₯<1,βˆ’4π‘₯+7π‘₯π‘₯β‰₯1.forfor

  • A ( 0 , 4 )
  • B ( 0 , 0 )
  • C 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 .
  • D ( 0 , βˆ’ 7 )

Q18:

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

Q19:

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

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

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

Consider the function 𝑔(π‘₯)=π‘₯οŠͺ. Is 𝑔′ increasing on the interval (βˆ’1,1)?

  • Ano
  • Byes

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

  • Ano
  • Byes

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

  • Ano
  • Byes

Q20:

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

  • AThe function is concave up on the intervals (βˆ’βˆž,βˆ’1) and (1,∞) and down on the interval (βˆ’1,1).
  • BThe function is concave up on the interval (1,∞) and down on the intervals (βˆ’βˆž,βˆ’1) and (βˆ’1,1) .
  • CThe function is concave up on the interval (βˆ’1,1) and down on the intervals (βˆ’βˆž,βˆ’1) and (1,∞).
  • DThe function is concave up on the intervals (βˆ’1,1) and (1,∞) and down on the interval (βˆ’βˆž,βˆ’1).
  • EThe function is concave up on the intervals (βˆ’βˆž,βˆ’1) and (βˆ’1,1) and down on the interval (1,∞).

Q21:

Find the intervals over which the graph of the function 𝑓(π‘₯)=2π‘₯βˆ’5π‘₯+11οŠͺ is convex downwards and convex upwards.

  • Aconvex downwards over the intervals ο€Ώβˆ’βˆž,βˆ’βˆš156 and ο€Ώβˆš156,βˆžο‹, convex upwards over the interval ο€Ώβˆ’βˆš156,√156
  • Bconvex upwards over the intervals ο€Ώβˆ’βˆž,βˆ’βˆš156 and ο€Ώβˆ’βˆš156,√156, convex downwards over the interval ο€Ώβˆš156,βˆžο‹
  • Cconvex downwards over the interval ο€Ώβˆ’βˆž,√156, convex upwards over the interval ο€Ώβˆš156,βˆžο‹
  • Dconvex upwards over the interval ο€Ώβˆ’βˆž,√156, convex downwards over the interval ο€Ώβˆš156,βˆžο‹

Q22:

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

  • AThe inflection point is (5,βˆ’22).
  • BThe inflection point is (3,βˆ’14).
  • CThe inflection point is (4,βˆ’20).
  • DThe inflection point is (4,βˆ’18).
  • EThe inflection point is (4,18).

Q23:

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

  • Ainflection points at ο€Ώβˆš33,βˆ’409 and ο€Ώβˆ’βˆš33,βˆ’409
  • Binflection points at (1,βˆ’4), (βˆ’1,βˆ’4) and (0,5)
  • Cinflection points at ο€Ώβˆš33,409 and ο€Ώβˆ’βˆš33,409
  • Dinflection points at (1,4), (βˆ’1,4) and (0,5)
  • Einflection points at ο€Ώβˆš33,409 and ο€Ώβˆ’βˆš33,βˆ’409

Q24:

Find the intervals over which the graph of the function 𝑓(π‘₯)=π‘₯βˆ’3π‘₯βˆ’7π‘₯ is convex downwards and convex upwards.

  • AThe graph is convex downwards over the interval (βˆ’βˆž,1), and is convex downwards over the interval (1,∞).
  • BThe graph is convex upwards over the interval (βˆ’βˆž,1), and is convex upwards over the interval (1,∞).
  • CThe graph is convex downwards over the interval (βˆ’βˆž,1), and is convex upwards over the interval (1,∞).
  • DThe graph is convex upwards over the interval (βˆ’βˆž,1), and is convex downwards over the interval (1,∞).

Q25:

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

  • AThe inflection point are ο€Ό1,βˆ’52 and ο€Όβˆ’1,βˆ’52.
  • BThe inflection point are (1,βˆ’4) and (βˆ’1,4).
  • CThe inflection points are ο€Ό1,32 and ο€Όβˆ’1,32.
  • DThe inflection point are ο€Ό1,112 and ο€Όβˆ’1,112.
  • EThe inflection point are (1,βˆ’2) and (βˆ’1,2).

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