# 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.

**Q3: **

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

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

**Q4: **

Determine the intervals on which is concave up and down.

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

**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 and concave up on .
- CThe function is concave down on and concave up on and .
- DThe function is concave down on and concave up on and .
- 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 interval and concave down on the intervals and .
- C is concave up on the intervals and and concave down on the interval .
- D is concave up on the intervals and and concave down on the interval .
- 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 upward on and concave downward on .

**Q10: **

Find the inflection points of .

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

**Q11: **

Find the inflection point on the curve .

- Ahas no inflection points
- B
- C
- D
- E

**Q13: **

For , find all the inflection points of .

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

**Q14: **

Find the inflection points of .

- AThe inflection points are and .
- BThe inflection point is .
- CThe inflection point is .
- DThe inflection points are and .
- EThe inflection points are and .

**Q15: **

Find (if any) the inflection points of .

- AThe inflection point is .
- BThe inflection point is .
- CThere are no inflection points.
- DThe inflection point is .
- EThe inflection point is .

**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 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 . Is increasing on the interval ?

- Ano
- Byes

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

- Ano
- Byes

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

- Ano
- Byes

**Q20: **

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

- AThe function is concave up on the intervals and and down on the interval .
- BThe function is concave up on the interval and down on the intervals and .
- CThe function is concave up on the interval and down on the intervals and .
- DThe function is concave up on the intervals and and down on the interval .
- EThe 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 downwards over the intervals and , convex upwards over the interval
- Bconvex upwards over the intervals and , convex downwards over the interval
- Cconvex downwards over the interval , convex upwards over the interval
- Dconvex upwards over the interval , convex downwards over the interval

**Q22: **

Find the inflection point of the function .

- AThe inflection point is .
- BThe inflection point is .
- CThe inflection point is .
- DThe inflection point is .
- EThe 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.

- AThe graph is convex downwards over the interval , and is convex downwards over the interval .
- BThe graph is convex upwards over the interval , and is convex upwards over the interval .
- CThe graph is convex downwards over the interval , and is convex upwards over the interval .
- DThe graph is convex upwards over the interval , and is convex downwards over the interval .