Lesson: Concavity and Points of Inflection

In this lesson, we will learn how to determine the concavity of functions as well as their inflection points.

Sample Question Videos

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Worksheet: Concavity and Points of Inflection • 25 Questions • 3 Videos

Q1:

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

Q2:

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

Q3:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ + π‘₯ 5 3 is concave up and down.

Q4:

Determine the intervals on which is concave up and down.

Q5:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = βˆ’ 3 π‘₯ + √ 9 π‘₯ + 1 2 is concave up and down.

Q6:

Determine the intervals on which the function is concave up and down.

Q7:

For , determine the intervals on which is concave up and concave down.

Q8:

Determine the intervals on which the function is concave upward and downward.

Q9:

Find the inflection point on the graph of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 9 π‘₯ + 6 π‘₯ 3 2 .

Q10:

Find the inflection points of 𝑓 ( π‘₯ ) = βˆ’ 2 π‘₯ + 5 π‘₯ 5 3 .

Q11:

Find the inflection point on the curve 𝑦 = 6 π‘₯ ( π‘₯ + 1 ) 2 .

Q12:

Given that 𝑓 ( π‘₯ ) = 4 π‘₯ + 4 π‘₯ s i n c o s , where 0 ≀ π‘₯ ≀ πœ‹ 2 , determine the inflection points of 𝑓 .

Q13:

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

Q14:

Find the inflection points of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 4 π‘₯ + 1 2 2 .

Q15:

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

Q16:

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

Q17:

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

Q18:

The curve 𝑦 = π‘˜ π‘₯ + π‘₯ βˆ’ 5 2 3 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 𝑓 β€² is increasing on the interval. So the slope of the graph gets larger as π‘₯ increases.

If 𝑓 β€² β€² exists on the interval, what result would prove that 𝑓 is concave up if 𝑓 β€² β€² ( π‘₯ ) > 0 for π‘₯ in 𝐽 ?

Consider the function 𝑔 ( π‘₯ ) = π‘₯ 4 . Is 𝑔 β€² increasing on the interval ( βˆ’ 1 , 1 ) ?

With the function above, our definition says that the function 𝑔 is concave up on ( βˆ’ 1 , 1 ) . Is 𝑔 β€² β€² ( π‘₯ ) > 0 on this interval?

Is it true that if 𝑓 is concave up on an interval, then 𝑓 β€² β€² ( π‘₯ ) > 0 on the interval? (Recall the definition above!)

Q20:

Determine where is concave up and where it is concave down.

Q21:

Find the intervals over which the graph of the function 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 5 π‘₯ + 1 1 4 2 is convex downwards and convex upwards.

Q22:

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

Q23:

Find all the inflection points of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 2 π‘₯ + 5 4 2 .

Q24:

Find the intervals over which the graph of the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 3 π‘₯ βˆ’ 7 π‘₯ 3 2 is convex downwards and convex upwards.

Q25:

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

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