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Worksheet: Concavity and Points of Inflection

Q1:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 4 𝑥 + 𝑥 5 3 is concave up and down.

  • A The function is concave up on , 3 0 2 0 and 3 0 2 0 , 0 and concave down on 0 , 3 0 2 0 and 3 0 2 0 , .
  • B The function is concave up on 3 0 2 0 , 0 and 3 0 2 0 , and concave down on , 3 0 2 0 and 0 , 3 0 2 0 .
  • C The function is concave up on 0 , 3 0 2 0 and 3 0 2 0 , and concave down on , 3 0 2 0 and 3 0 2 0 , 0 .
  • D The function is concave up on , 3 0 2 0 and 0 , 3 0 2 0 and concave down on 3 0 2 0 , 0 and 3 0 2 0 , .
  • E The function is concave up on 3 0 2 0 , 0 and 0 , 3 0 2 0 and concave down on , 3 0 2 0 and 3 0 2 0 , .

Q2:

Determine the intervals on which 𝑓 ( 𝑥 ) = 4 𝑥 + ( 𝑥 + 3 ) 4 5 is concave up and down.

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

Q3:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 3 𝑥 + 9 𝑥 + 1 2 is concave up and down.

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

Q4:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 𝑥 3 2 𝑥 + 1 2 2 is concave up and down.

  • AThe function is concave down on , 6 6 and 6 6 , 6 6 and concave up on 6 6 , .
  • BThe function is concave down on 6 6 , 6 6 and concave up on , 6 6 and 6 6 , .
  • CThe function is concave down on 6 6 , and concave up on , 6 6 and 6 6 , 6 6 .
  • DThe function is concave down on , 6 6 and 6 6 , and concave up on 6 6 , 6 6 .
  • EThe function is concave down on 6 6 , 6 6 and 6 6 , and concave up on , 6 6 .

Q5:

For 0 < 𝑥 < 2 𝜋 , determine the intervals on which 𝑓 ( 𝑥 ) = 𝑥 2 𝑥 c o s s i n 2 is concave up and concave down.

  • A 𝑓 is concave up on the interval 𝜋 6 , 5 𝜋 6 and concave down on the intervals 0 , 𝜋 2 and 3 𝜋 2 , 2 𝜋 .
  • B 𝑓 is concave up on the intervals 0 , 𝜋 6 and 5 𝜋 6 , 2 𝜋 and concave down on the interval 𝜋 6 , 5 𝜋 6 .
  • C 𝑓 is concave up on the intervals 0 , 𝜋 2 and 3 𝜋 2 , 2 𝜋 and concave down on the interval 𝜋 6 , 5 𝜋 6 .
  • D 𝑓 is concave up on the interval 𝜋 6 , 5 𝜋 6 and concave down on the intervals 0 , 𝜋 6 and 5 𝜋 6 , 2 𝜋 .
  • E 𝑓 is concave up on the interval ( 𝜋 , 2 𝜋 ) and concave down on the interval ( 0 , 𝜋 ) .

Q6:

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

  • AThe function is concave upward on 2 𝑒 , and concave downward on 0 , 2 𝑒 .
  • BThe function is concave upward on 0 , 𝑒 2 3 2 and concave downward on 𝑒 2 , 3 2 .
  • CThe function is concave upward on 1 2 𝑒 , and concave downward on 0 , 1 2 𝑒 .
  • DThe function is concave upward on 𝑒 2 , 3 2 and concave downward on 0 , 𝑒 2 3 2 .
  • EThe function is concave downward on 0 , 1 2 𝑒 and concave downward on 1 2 𝑒 , .

Q7:

Find the inflection point on the graph of 𝑓 ( 𝑥 ) = 𝑥 9 𝑥 + 6 𝑥 3 2 .

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

Q8:

Find the inflection points of 𝑓 ( 𝑥 ) = 2 𝑥 + 5 𝑥 5 3 .

  • A 3 2 , 4 5 8 , 3 2 , 4 5 8 , ( 0 , 3 ) .
  • B 3 2 , 3 3 3 2 , 3 2 , 3 3 3 2 .
  • C 3 2 , 4 5 8 , 3 2 , 4 5 8 .
  • D 3 2 , 2 1 3 1 6 , 3 2 , 2 1 3 1 6 , ( 0 , 0 ) .
  • E 5 6 6 , 4 7 5 3 6 , 5 6 6 , 4 7 5 3 6 , ( 0 , 3 ) .

Q9:

Find the inflection point on the curve 𝑦 = 6 𝑥 ( 𝑥 + 1 ) 2 .

  • A 1 3 , 8 9
  • B 2 3 , 1 0 0 9
  • C 3 2 , 9 4
  • D 2 3 , 4 9
  • Ehas no inflection points

Q10:

Find the inflection point on the curve 𝑦 = 9 𝑥 ( 𝑥 1 ) 2 .

  • A 1 3 , 4 3
  • B 2 3 , 5 0 3
  • C 3 2 , 2 7 8
  • D 2 3 , 2 3
  • Ehas no inflection points

Q11:

Given that 𝑓 ( 𝑥 ) = 4 𝑥 + 4 𝑥 s i n c o s , where 0 𝑥 𝜋 2 , determine the inflection points of 𝑓 .

  • A 𝑓 has inflection points at 3 𝜋 1 6 , 2 and 7 𝜋 1 6 , 2 .
  • B 𝑓 has inflection points at 3 𝜋 1 6 , 2 and 7 𝜋 1 6 , 2 .
  • C 𝑓 has inflection points at 𝜋 1 6 , 2 and 5 𝜋 1 6 , 2 .
  • D 𝑓 has inflection points at 3 𝜋 1 6 , 0 and 7 𝜋 1 6 , 0 .
  • E 𝑓 has inflection points at 𝜋 1 6 , 0 and 5 𝜋 1 6 , 0 .

Q12:

For 0 𝑥 4 𝜋 , find all the inflection points of 𝑓 ( 𝑥 ) = 2 𝑥 𝑥 s i n .

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

Q13:

Find the inflection points of 𝑓 ( 𝑥 ) = 𝑥 1 4 𝑥 + 1 2 2 .

  • A The inflection point is 1 4 , 3 4 .
  • B The inflection points are 1 2 , 3 8 and 1 2 , 3 8 .
  • C The inflection point is 4 , 3 1 3 .
  • D The inflection points are 3 6 , 1 1 1 6 and 3 6 , 1 1 1 6 .
  • E The inflection points are 3 2 , 1 1 6 and 3 2 , 1 1 6 .

Q14:

Find (if any) the inflection points of 𝑓 ( 𝑥 ) = 𝑒 2 𝑒 + 5 𝑥 𝑥 .

  • AThe inflection point is 0 , 1 7 .
  • BThe inflection point is 0 , 1 3 .
  • CThe inflection point is 1 5 , 𝑒 2 𝑒 + 5 5 5 .
  • DThere are no inflection points.
  • EThe inflection point is 1 5 , 𝑒 2 𝑒 + 5 5 5 .

Q15:

Find (if any) the inflection points of 𝑓 ( 𝑥 ) = 3 𝑥 2 𝑥 2 l n .

  • A 𝑓 has an inflection point at 1 2 𝑒 , 3 8 𝑒 .
  • B 𝑓 has an inflection point at 𝑒 2 , 9 8 𝑒 3 3 2 .
  • C 𝑓 has an inflection point at 1 2 𝑒 , 3 8 𝑒 .
  • D 𝑓 has an inflection point at 𝑒 2 , 9 8 𝑒 3 3 2 .
  • E 𝑓 has no inflection points.

Q16:

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

  • A ( 0 , 7 )
  • B T h e f u n c t i o n h a s n o i n e c t i o n p o i n t s .
  • C ( 0 , 0 )
  • D ( 0 , 4 )

Q17:

The curve 𝑦 = 𝑘 𝑥 + 𝑥 5 2 3 has an inflection point at 𝑥 = 1 . What is 𝑘 ?

Q18:

A good definition of the function 𝑓 being concave up on an interval 𝐽 = ( 𝑎 , 𝑏 ) is that 𝑓 is increasing on the interval. So the slope of the graph gets larger as 𝑥 increases.

If 𝑓 exists on the interval, what result would prove that 𝑓 is concave up if 𝑓 ( 𝑥 ) > 0 for 𝑥 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 ( 𝑥 ) = 𝑥 4 . Is 𝑔 increasing on the interval ( 1 , 1 ) ?

  • Ayes
  • Bno

With the function above, our definition says that the function 𝑔 is concave up on ( 1 , 1 ) . Is 𝑔 ( 𝑥 ) > 0 on this interval?

  • Ayes
  • Bno

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

  • Ayes
  • Bno

Q19:

Determine where 𝑓 ( 𝑥 ) = 𝑥 2 3 𝑥 + 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 ( , 1 ) and ( 1 , 1 ) .
  • B The function is concave up on the interval ( 1 , 1 ) and down on the intervals ( , 1 ) and ( 1 , ) .
  • C The function is concave up on the intervals ( , 1 ) and ( 1 , 1 ) and down on the interval ( 1 , ) .
  • D The function is concave up on the intervals ( , 1 ) and ( 1 , ) and down on the interval ( 1 , 1 ) .
  • E The function is concave up on the intervals ( 1 , 1 ) and ( 1 , ) and down on the interval ( , 1 ) .

Q20:

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

Q21:

Find the inflection point of the function 𝑓 ( 𝑥 ) = 5 𝑥 + ( 𝑥 4 ) + 2 5 .

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

Q22:

Find all the inflection points of 𝑓 ( 𝑥 ) = 𝑥 2 𝑥 + 5 4 2 .

  • Ainflection points at 3 3 , 4 0 9 and 3 3 , 4 0 9
  • Binflection points at 3 3 , 4 0 9 and 3 3 , 4 0 9
  • Cinflection points at ( 1 , 4 ) , ( 1 , 4 ) and ( 0 , 5 )
  • Dinflection points at 3 3 , 4 0 9 and 3 3 , 4 0 9
  • Einflection points at ( 1 , 4 ) , ( 1 , 4 ) and ( 0 , 5 )

Q23:

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 .

Q24:

Find the inflection points of the function 𝑓 ( 𝑥 ) = 𝑥 2 3 𝑥 + 4 4 2 .

  • A The inflection point are 1 , 5 2 and 1 , 5 2 .
  • B The inflection point are ( 1 , 4 ) and ( 1 , 4 ) .
  • C The inflection point are ( 1 , 2 ) and ( 1 , 2 ) .
  • D The inflection points are 1 , 3 2 and 1 , 3 2 .
  • E The inflection point are 1 , 1 1 2 and 1 , 1 1 2 .

Q25:

Find the inflection point of 𝑓 ( 𝑥 ) = 𝑥 1 2 𝑥 1 3 .

  • AThe inflection point is 2 3 , 1 .
  • BThe inflection point is ( 2 , 1 7 ) .
  • CThe inflection point is ( 0 , 1 2 ) .
  • DThe inflection point is ( 0 , 1 ) .
  • EThe inflection point is ( 0 , 2 ) .

Q26:

Find the inflection point of the function 𝑓 ( 𝑥 ) = 4 𝑥 2 𝑥 + 1 6 𝑥 3 2 .

  • A 1 6 , 4 9 3
  • B 1 6 , 7 0 2 7
  • C 1 6 , 1 4 6 9
  • D 1 6 , 7 3 2 7
  • E 1 6 , 1 4 0 9

Q27:

Let 𝑓 ( 𝑥 ) = 𝑥 2 𝑥 c o s s i n 2 . Find all the inflection points of 𝑓 with 0 < 𝑥 < 2 𝜋 .

  • A 𝑓 has inflection points at 𝜋 6 , 7 4 and 5 𝜋 6 , 1 4 .
  • B 𝑓 has inflection points at 𝜋 6 , 1 4 and 5 𝜋 6 , 1 4 .
  • C 𝑓 has inflection points at 3 𝜋 2 , 2 and 𝜋 2 , 2 .
  • D 𝑓 has inflection points at 𝜋 6 , 1 4 and 5 𝜋 6 , 1 4 .
  • E 𝑓 has inflection points at 3 𝜋 2 , 2 and 𝜋 2 , 2 .

Q28:

Determine the intervals where the graph of is convex up and where it is convex down.

  • A downwards over the interval
  • B downwards over the interval , upwards over the intervals and
  • C upwards over the intervals and
  • D upwards over the interval , downwards over the intervals and

Q29:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 1 + 3 𝑥 5 𝑥 2 is concave up and down.

  • A The function is concave down on the intervals ( , 0 ) and 0 , 1 0 3 and up on the interval 1 0 3 , .
  • B The function is concave down on the interval ( 5 , ) and up on the intervals ( , 0 ) and ( 0 , 5 ) .
  • C The function is concave down on the interval 1 0 3 , and up on the intervals ( , 0 ) and 0 , 1 0 3 .
  • D The function is concave down on the interval ( 0 , 5 ) and up on the interval ( 5 , ) .
  • E The function is concave down on the intervals ( , 0 ) and ( 5 , ) and up on the interval ( 0 , 5 ) .

Q30:

Find the inflection points, if any, of the function 𝑓 ( 𝑥 ) = 𝑥 4 9 𝑥 3 6 2 2 .

  • A ( 7 , 0 )
  • B ( 7 , 0 )
  • C ( 7 , 0 ) , ( 7 , 0 )
  • D The function has no inflection points.

Q31:

Let 𝑓 ( 𝑥 ) = 2 𝑥 𝑥 + 4 2 . Determine the intervals where this function is concave up/down.

  • Aconcaves up on , 2 3 and 2 3 , 0 , concave down on 0 , 2 3 and 2 3 , .
  • Bconcaves up on , 2 3 and 0 , 2 3 , concave down on 2 3 , 0 and 2 3 , .
  • Cconcaves up on ( 2 , 0 ) and ( 2 , ) , concave down on ( , 2 ) and ( 0 , 2 ) .
  • Dconcaves up on 2 3 , 0 and 2 3 , , concave down on , 2 3 and 0 , 2 3 .
  • Econcaves up on ( , 2 ) and ( 0 , 2 ) , concave down on ( 2 , 0 ) and ( 2 , ) .

Q32:

Determine the inflection point of the function 𝑓 ( 𝑥 ) = 4 + 5 𝑥 2 𝑥 2 .

  • A 6 5 , 8 6 9
  • B 4 5 , 7 8
  • C 5 6 , 2 2 2 5
  • D 6 5 , 1 1 9
  • E 5 4 , 3 2 2 5

Q33:

Find, if any, the inflection points on the graph of 𝑓 ( 𝑥 ) = 8 2 𝑥 + 1 2 .

  • A 6 , 8 1 3 6 , 8 1 3 ,
  • B ( 0 , 8 )
  • C The function has no inflection points.
  • D 6 6 , 6 6 6 , 6 ,

Q34:

Determine the intervals where the graph of is convex up and where it is convex down.

  • Aupwards on
  • Bupwards on , downwards on ,
  • Cdownwards on ,
  • Ddownwards on , upwards on ,

Q35:

Find (if any) the inflection points of 𝑓 ( 𝑥 ) = 5 𝑥 𝑥 + 4 2 .

  • A 𝑓 has inflection points at 2 , 5 4 , ( 0 , 0 ) , and 2 , 5 4 .
  • B 𝑓 has inflection points at 2 3 , 5 3 8 and 2 3 , 5 3 8 .
  • C 𝑓 has inflection points at 2 , 5 4 and 2 , 5 4 .
  • D 𝑓 has inflection points at 2 3 , 5 3 8 , ( 0 , 0 ) , and 2 3 , 5 3 8 .
  • E 𝑓 has inflection points at 2 3 , 5 3 8 , 2 , 5 4 , ( 0 , 0 ) , 2 , 5 4 , and 2 3 , 5 3 8 .

Q36:

Find, if any, the inflection points of 𝑦 = 2 𝑥 6 𝑥 1 .

  • A ( 2 . 7 3 , 8 . 9 3 )
  • B ( 0 . 7 3 , 4 . 9 3 )
  • C ( 0 . 7 3 , 4 . 9 3 ) , ( 2 . 7 3 , 8 . 9 3 )
  • DThe function has no inflection points.

Q37:

Determine the inflection points of the curve 𝑓 ( 𝑥 ) = 7 𝑥 𝑥 2 2 2 .

  • A ( 2 2 , 1 1 )
  • B ( 0 , 0 ) , ( 4 4 , 8 8 )
  • C ( 0 , 0 ) , ( 4 4 , 8 8 )
  • Dhas no inflection points

Q38:

The figure shows the graph of 𝑓 ( 𝑥 ) = 𝐴 𝑥 + 𝐵 𝑥 2 3 for positive constants 𝐴 , 𝐵 .

Find the exact values of the constants if the local minimum value shown is 1 and the inflection point 𝑃 occurs when 𝑥 = 2 .

  • A 𝐴 = 9 4 , 𝐵 = 9 4
  • B 𝐴 = 3 2 , 𝐵 = 3 2
  • C 𝐴 = 4 2 7 , 𝐵 = 4 2 7
  • D 𝐴 = 2 7 4 , 𝐵 = 2 7 4
  • E 𝐴 = 4 9 , 𝐵 = 4 9

Q39:

Find the inflection points of 𝑓 ( 𝑥 ) = 3 𝑥 + 9 𝑥 + 5 2 .

  • A The inflection point is 9 5 , 2 7 5 + 8 5 4 5 .
  • B The inflection point is 0 , 3 + 5 .
  • C The inflection point is 5 9 , 5 3 + 7 0 3 .
  • D There are no inflection points.
  • E The inflection point is 9 5 , 2 7 5 + 8 5 4 5 .

Q40:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 4 𝑥 ( 3 𝑥 + 4 ) 1 3 is concave up, and the intervals on which it is concave down.

  • A The function is concave up on the intervals ( , 0 ) and 0 , 2 3 and down on the interval 2 3 , .
  • B The function is concave up on the interval 0 , 2 3 and down on the intervals ( , 0 ) and 2 3 , .
  • C The function is concave up on the interval 2 3 , and down on the intervals ( , 0 ) and 0 , 2 3 .
  • D The function is concave up on the intervals ( , 0 ) and 2 3 , and down on the interval 0 , 2 3 .
  • E The function is concave up on the intervals 0 , 2 3 and 2 3 , and down on the interval ( , 0 ) .

Q41:

Find the inflection points of the function 𝑓 ( 𝑥 ) = 𝑥 3 𝑥 + 2 .

  • A 4 9 , 4 3 0 2 7
  • B 4 9 , 4 6 2 7
  • C 2 3 , 0
  • Dno inflection points
  • E 8 9 , 8 4 2 2 7

Q42:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 𝑒 + 𝑒 5 𝑥 𝑥 is concave upward and downward.

  • AThe function is concave upward on , 5 6 l n and concave downward on 5 6 , l n .
  • BThe function is concave upward on 5 6 , l n and concave downward on , 5 6 l n .
  • CThe function is concave upward on 5 6 , l n and concave downward on , 5 6 l n .
  • DThe function is concave upward on ( , ) .
  • EThe function is concave downward on ( , ) .

Q43:

Find (if any) the inflection points of 𝑓 ( 𝑥 ) = 𝑒 + 𝑒 6 𝑥 2 𝑥 .

  • Ainflection point at 3 4 , 3 + 3 l n 3 2 1 2
  • Binflection point at 3 8 , 3 + 3 l n 3 4 1 4
  • Cinflection point at 3 8 , 3 2 l n
  • DThe function has no inflection points.
  • Einflection point at 3 4 , 3 l n

Q44:

Given that 𝑓 ( 𝑥 ) = 2 𝑥 𝑒 4 4 𝑥 , find all the inflection points of 𝑓 .

  • A 𝑓 has inflection points at 3 2 , 8 1 8 𝑒 6 and 1 2 , 1 8 𝑒 2 .
  • B 𝑓 has inflection points at ( 0 , 0 ) and 1 , 1 0 𝑒 4 .
  • C 𝑓 has inflection points at 1 , 2 𝑒 4 and ( 0 , 0 ) .
  • D 𝑓 has inflection points at 1 2 , 1 8 𝑒 2 and 3 2 , 8 1 8 𝑒 6 .
  • E 𝑓 has no inflection points.

Q45:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 4 𝑒 2 𝑥 2 is concave up and down.

  • AThe function is concave up on , 1 2 and 1 2 , 1 2 and concave down on 1 2 , .
  • BThe function is concave up on 1 2 , 1 2 and concave down on , 1 2 and 1 2 , .
  • CThe function is concave up on 1 2 , and concave down on , 1 2 and 1 2 , 1 2 .
  • DThe function is concave up on , 1 2 and 1 2 , and concave down on 1 2 , 1 2 .
  • EThe function is concave up on , 1 2 and concave down on 1 2 , 1 2 and 1 2 ,