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=𝑥+2𝑥5.

  • 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)=4𝑥33𝑥+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)=𝑥2𝑥cossin 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 0,12𝑒 and concave downward on 12𝑒,.
  • CThe function is concave upward on 2𝑒, and concave downward on 0,2𝑒.
  • DThe function is concave upward on 12𝑒, and concave downward on 0,12𝑒.
  • EThe function is concave upward on 0,𝑒2 and concave downward on 𝑒2,.

Q9:

Find the inflection point on the graph of 𝑓(𝑥)=𝑥9𝑥+6𝑥.

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

Q10:

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

  • A32,458, 32,458, (0,3).
  • B32,458, 32,458 .
  • C32,3332, 32,3332.
  • D566,47536, 566,47536, (0,3).
  • E32,21316, 32,21316, (0,0).

Q11:

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

  • Ahas no inflection points
  • B32,94
  • C23,1009
  • D23,49
  • E13,89

Q12:

Given that 𝑓(𝑥)=4𝑥+4𝑥sincos, where 0𝑥𝜋2, 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 0𝑥4𝜋, find all the inflection points of 𝑓(𝑥)=2𝑥𝑥sin.

  • 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)=𝑥14𝑥+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)=𝑒2𝑒+5.

  • AThe inflection point is 0,17.
  • BThe inflection point is 0,13.
  • CThere are no inflection points.
  • DThe inflection point is 15,𝑒2𝑒+5.
  • EThe inflection point is 15,𝑒2𝑒+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 12𝑒,38𝑒.
  • E𝑓 has an inflection point at 12𝑒,38𝑒.

Q17:

Find, if any, the inflection points on the graph of 𝑓(𝑥)=6𝑥7𝑥+4𝑥<1,4𝑥+7𝑥𝑥1.forfor

  • A(0,4)
  • B(0,0)
  • CThefunctionhasnoinectionpoints.
  • D(0,7)

Q18:

The curve 𝑦=𝑘𝑥+𝑥5 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)=𝑥23𝑥+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)=2𝑥5𝑥+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+(𝑥4)+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)=𝑥2𝑥+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)=𝑥3𝑥7𝑥 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)=𝑥23𝑥+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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