Worksheet: Concavity and Points of Inflection

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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=𝑥+2𝑥−5.

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

Q3:

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

  • AThe function is concave up on −√3020,0 and 0,√3020 and concave down on −∞,−√3020 and √3020,∞ .
  • BThe function is concave up on −∞,−√3020 and 0,√3020 and concave down on −√3020,0 and √3020,∞.
  • CThe function is concave up on −√3020,0 and √3020,∞ and concave down on −∞,−√3020 and 0,√3020.
  • DThe function is concave up on 0,√3020 and √3020,∞ and concave down on −∞,−√3020 and −√3020,0.
  • EThe function is concave up on −∞,−√3020 and −√3020,0 and concave down on 0,√3020 and √3020,∞ .

Q4:

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

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

Q5:

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

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

Q6:

Determine the intervals on which the function 𝑓(𝑥)=4𝑥−33𝑥+1 is concave up and down.

  • AThe function is concave down on −∞,−13 and −13,13 and concave up on 13,∞.
  • BThe function is concave down on −13,13 and 13,∞ and concave up on −∞,−13.
  • CThe function is concave down on −13,13 and concave up on −∞,−13 and 13,∞.
  • DThe function is concave down on 13,∞ and concave up on −∞,−13 and −13,13.
  • EThe function is concave down on −∞,−13 and 13,∞ and concave up on −13,13.

Q7:

For 0<𝑥<2𝜋, determine the intervals on which 𝑓(𝑥)=𝑥−2𝑥cossin is concave up and concave down.

  • A𝑓 is concave up on the interval 𝜋6,5𝜋6 and concave down on the intervals 0,𝜋6 and 5𝜋6,2𝜋.
  • B𝑓 is concave up on the interval 𝜋6,5𝜋6 and concave down on the intervals 0,𝜋2 and 3𝜋2,2𝜋.
  • C𝑓 is concave up on the intervals 0,𝜋2 and 3𝜋2,2𝜋 and concave down on the interval 𝜋6,5𝜋6.
  • D𝑓 is concave up on the intervals 0,𝜋6 and 5𝜋6,2𝜋 and concave down on the interval 𝜋6,5𝜋6.
  • E𝑓 is concave up on the interval (𝜋,2𝜋) and concave down on the interval (0,𝜋).

Q8:

Determine the intervals on which the function 𝑓(𝑥)=3𝑥2𝑥ln is concave upward and downward.

  • AThe function is concave upward on 𝑒2,∞ and concave downward on 0,𝑒2.
  • BThe function is concave upward on 0,12√𝑒 and concave downward on 12√𝑒,∞.
  • CThe function is concave upward on 2√𝑒,∞ and concave downward on 0,2√𝑒.
  • DThe function is concave upward on 12√𝑒,∞ and concave downward on 0,12√𝑒.
  • EThe function is concave upward on 0,𝑒2 and concave downward on 𝑒2,∞.

Q9:

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

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

Q10:

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

  • A√32,458, −√32,−458, (0,3).
  • B√32,458, −√32,−458 .
  • C√32,−3√332, −√32,3√332.
  • D5√66,47536, −5√66,−47536, (0,−3).
  • E√32,21√316, −√32,−21√316, (0,0).

Q11:

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

  • Ahas no inflection points
  • B−32,−94
  • C23,1009
  • D−23,−49
  • E−13,−89

Q12:

Given that 𝑓(𝑥)=4𝑥+4𝑥sincos, where 0≤𝑥≤𝜋2, determine the inflection points of 𝑓.

  • A𝑓 has inflection points at 3𝜋16,√2 and 7𝜋16,−√2.
  • B𝑓 has inflection points at 3𝜋16,√2 and 7𝜋16,√2.
  • C𝑓 has inflection points at 3𝜋16,0 and 7𝜋16,0.
  • D𝑓 has inflection points at 𝜋16,√2 and 5𝜋16,−√2.
  • E𝑓 has inflection points at 𝜋16,0 and 5𝜋16,0.

Q13:

For 0≤𝑥≤4𝜋, find all the inflection points of 𝑓(𝑥)=2𝑥−𝑥sin.

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

Q14:

Find the inflection points of 𝑓(𝑥)=𝑥−14𝑥+1.

  • AThe inflection points are √32,−116 and −√32,−116.
  • BThe inflection point is 14,−34.
  • CThe inflection point is −4,313.
  • DThe inflection points are √36,−1116 and −√36,−1116.
  • EThe inflection points are 12,−38 and −12,−38.

Q15:

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

  • AThe inflection point is 0,17.
  • BThe inflection point is 0,13.
  • CThere are no inflection points.
  • DThe inflection point is 15,√𝑒2√𝑒+5.
  • EThe inflection point is 15,√𝑒−2√𝑒+5.

Q16:

Find (if any) the inflection points of 𝑓(𝑥)=3𝑥2𝑥ln.

  • A𝑓 has an inflection point at 𝑒2,98𝑒.
  • B𝑓 has no inflection points.
  • C𝑓 has an inflection point at 𝑒2,−98𝑒.
  • D𝑓 has an inflection point at 12√𝑒,−38𝑒.
  • E𝑓 has an inflection point at 12√𝑒,38𝑒.

Q17:

Find, if any, the inflection points on the graph of 𝑓(𝑥)=6𝑥−7𝑥+4𝑥<1,−4𝑥+7𝑥𝑥≥1.forfor

  • A(0,4)
  • B(0,0)
  • CThefunctionhasnoinflectionpoints.
  • D(0,−7)

Q18:

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

Q19:

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 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(𝑥)=𝑥. Is 𝑔′ increasing on the interval (−1,1)?

  • Ano
  • Byes

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

  • Ano
  • Byes

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

  • Ano
  • Byes

Q20:

Determine where 𝑓(𝑥)=𝑥2−3𝑥+3 is concave up and where it is concave down.

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

Q21:

Find the intervals over which the graph of the function 𝑓(𝑥)=2𝑥−5𝑥+11 is convex downwards and convex upwards.

  • Aconvex downwards over the intervals −∞,−√156 and √156,∞, convex upwards over the interval −√156,√156
  • Bconvex upwards over the intervals −∞,−√156 and −√156,√156, convex downwards over the interval √156,∞
  • Cconvex downwards over the interval −∞,√156, convex upwards over the interval √156,∞
  • Dconvex upwards over the interval −∞,√156, convex downwards over the interval √156,∞

Q22:

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

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

Q23:

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

  • Ainflection points at √33,−409 and −√33,−409
  • Binflection points at (1,−4), (−1,−4) and (0,5)
  • Cinflection points at √33,409 and −√33,409
  • Dinflection points at (1,4), (−1,4) and (0,5)
  • Einflection points at √33,409 and −√33,−409

Q24:

Find the intervals over which the graph of the function 𝑓(𝑥)=𝑥−3𝑥−7𝑥 is convex downwards and convex upwards.

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

Q25:

Find the inflection points of the function 𝑓(𝑥)=𝑥2−3𝑥+4.

  • AThe inflection point are 1,−52 and −1,−52.
  • BThe inflection point are (1,−4) and (−1,4).
  • CThe inflection points are 1,32 and −1,32.
  • DThe inflection point are 1,112 and −1,112.
  • EThe inflection point are (1,−2) and (−1,2).

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