# 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.

**Q2: **

Determine the inflection points of the curve .

- A
- B
- C
- Dhas no inflection points

**Q3: **

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

- A The function is concave up on and and concave down on and .
- B The function is concave up on and and concave down on and .
- C The function is concave up on and and concave down on and .
- D The function is concave up on and and concave down on and .
- E The function is concave up on and and concave down on and .

**Q4: **

Determine the intervals on which is concave up and down.

- A The function is concave down on the interval and up on the interval .
- B The function is concave down on the interval and up on the interval .
- C The function is concave down on the interval and up on the interval .
- D The function is concave down on the interval and up on the interval .
- E The function is concave down on the interval and up on the interval .

**Q5: **

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

- AThe function is concave up on .
- BThe function is concave down on .
- CThe function is concave down on .
- D The function is concave up on .
- EThe function is concave down on .

**Q6: **

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

- AThe function is concave down on and and concave up on .
- BThe function is concave down on and concave up on and .
- CThe function is concave down on and concave up on and .
- DThe function is concave down on and and concave up on .
- EThe function is concave down on and and concave up on .

**Q7: **

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

- A is concave up on the interval and concave down on the intervals and .
- B is concave up on the intervals and and concave down on the interval .
- C is concave up on the intervals and and concave down on the interval .
- D is concave up on the interval and concave down on the intervals and .
- E is concave up on the interval and concave down on the interval .

**Q8: **

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

- AThe function is concave upward on and concave downward on .
- BThe function is concave upward on and concave downward on .
- CThe function is concave upward on and concave downward on .
- DThe function is concave upward on and concave downward on .
- EThe function is concave downward on and concave downward on .

**Q9: **

Find the inflection point on the graph of .

- A
- Bno inflection point
- C
- D

**Q10: **

Find the inflection points of .

- A , , .
- B , .
- C , .
- D , , .
- E , , .

**Q11: **

Find the inflection point on the curve .

- A
- B
- C
- D
- Ehas no inflection points

**Q12: **

Given that , where , determine the inflection points of .

- A has inflection points at and .
- B has inflection points at and .
- C has inflection points at and .
- D has inflection points at and .
- E has inflection points at and .

**Q13: **

For , find all the inflection points of .

- A , ,
- B , ,
- C , ,
- D , ,
- E , ,

**Q14: **

Find the inflection points of .

- A The inflection point is .
- B The inflection points are and .
- C The inflection point is .
- D The inflection points are and .
- E The inflection points are and .

**Q15: **

Find (if any) the inflection points of .

- AThe inflection point is .
- BThe inflection point is .
- CThe inflection point is .
- DThere are no inflection points.
- EThe inflection point is .

**Q16: **

Find (if any) the inflection points of .

- A has an inflection point at .
- B has an inflection point at .
- C has an inflection point at .
- D has an inflection point at .
- E has no inflection points.

**Q17: **

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

- A
- B
- C
- D

**Q18: **

The curve has an inflection point at . What is ?

**Q19: **

A good definition of the function being concave up on an interval is that is increasing on the interval. So the slope of the graph gets larger as increases.

If exists on the interval, what result would prove that is concave up if for in ?

- 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 . Is increasing on the interval ?

- Ayes
- Bno

With the function above, our definition says that the function is concave up on . Is on this interval?

- Ayes
- Bno

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

- Ayes
- Bno

**Q20: **

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

- A The function is concave up on the interval and down on the intervals and .
- B The function is concave up on the interval and down on the intervals and .
- C The function is concave up on the intervals and and down on the interval .
- D The function is concave up on the intervals and and down on the interval .
- E The function is concave up on the intervals and and down on the interval .

**Q21: **

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

**Q22: **

Find the inflection point of the function .

- A The inflection point is .
- B The inflection point is .
- C The inflection point is .
- D The inflection point is .
- E The inflection point is .

**Q23: **

Find all the inflection points of .

- Ainflection points at and
- Binflection points at and
- Cinflection points at , and
- Dinflection points at and
- Einflection points at , and

**Q24: **

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 .

**Q25: **

Find the inflection points of the function .

- A The inflection point are and .
- B The inflection point are and .
- C The inflection point are and .
- D The inflection points are and .
- E The inflection point are and .