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Lesson Worksheet: Convexity and Points of Inflection Mathematics • Higher Education

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

Q1:

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‹ .

Q2:

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)

Q3:

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)

Q4:

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

  • A(3,βˆ’36)
  • B(3,βˆ’21)
  • C(3,0)
  • Dno inflection point

Q5:

The figure shows the graph of 𝑓(π‘₯)=βˆ’π΄π‘₯+𝐡π‘₯ for positive constants 𝐴,𝐡.

Find the exact values of the constants if the local minimum value shown is βˆ’1 and the inflection point 𝑃 occurs when π‘₯=2.

  • A𝐴=βˆ’49, 𝐡=βˆ’49
  • B𝐴=32, 𝐡=32
  • C𝐴=274, 𝐡=274
  • D𝐴=βˆ’94, 𝐡=βˆ’94
  • E𝐴=427, 𝐡=427

Q6:

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).

Q7:

Using the given graph of the function 𝑓, at what values of π‘₯ does 𝑓 have inflection points?

  • A𝑓 has inflection points when π‘₯=1 and π‘₯=7.
  • B𝑓 has inflection points when π‘₯=2, π‘₯=4, and π‘₯=6.
  • C𝑓 has inflection points when π‘₯=3 and π‘₯=5.
  • D𝑓 has inflection points when π‘₯=2 and π‘₯=6.
  • E𝑓 has inflection points when π‘₯=4 and π‘₯=6.

Q8:

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

  • A𝑓 has inflection points at ο€Ό3πœ‹16,√2 and ο€Ό7πœ‹16,βˆ’βˆš2.
  • B𝑓 has inflection points at ο€Ό3πœ‹16,√2 and ο€Ό7πœ‹16,√2.
  • C𝑓 has inflection points at ο€Ό3πœ‹16,0 and ο€Ό7πœ‹16,0.
  • D𝑓 has inflection points at ο€»πœ‹16,√2 and ο€Ό5πœ‹16,βˆ’βˆš2.
  • E𝑓 has inflection points at ο€»πœ‹16,0 and ο€Ό5πœ‹16,0.

Q9:

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

  • A𝑓 has an inflection point at 𝑒2,98π‘’οοŠ±οŠ©οŽ’οŽ‘.
  • B𝑓 has no inflection points.
  • C𝑓 has an inflection point at 𝑒2,βˆ’98π‘’οοŠ±οŠ©οŽ’οŽ‘.
  • D𝑓 has an inflection point at ο€Ώ12βˆšπ‘’,βˆ’38𝑒.
  • E𝑓 has an inflection point at ο€Ώ12βˆšπ‘’,38𝑒.

Q10:

Consider the parameric curve π‘₯=πœƒcos and 𝑦=πœƒsin. Determine whether this curve is concave up, down, or neither at πœƒ=πœ‹6.

  • Aupward
  • Bdownward
  • Cneither

This lesson includes 90 additional questions and 543 additional question variations for subscribers.

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