# Lesson Worksheet: Critical Points and Local Extrema of a Function Mathematics

In this worksheet, we will practice finding critical points of a function and checking for local extrema using the first derivative test.

**Q1: **

Determine the number of critical points of the following graph.

**Q2: **

Which graph has three real zeros and two local maxima?

- A
- B
- C

**Q3: **

Determine, if any, the local maximum and minimum values of , together with where they occur.

- AThe local maximum is at , and there is no local minimum.
- BThe local maximum is at , and the local minimum is at .
- CThe local minimum is at , and there is no local maximum.
- DThe local minimum is at , and the local maximum is at .

**Q4: **

Determine where the local maxima and minima are for .

- Alocal maxima at , local minima at and
- Blocal maxima at and , local minima at
- Clocal maxima at , local minima at and
- Dlocal maxima at and , local minima at
- Elocal maxima at and , local minima at

**Q5: **

Find the critical points of .

- AThe function has critical points at , , and .
- BThe function has critical points at , , and .
- CThe function has no critical points.
- DThe function has critical points at , , and .
- EThe function has critical points at , , and .

**Q6: **

The graph of the first derivative of a continuous function is shown. At what values of does have a local maximum and a local minimum?

- A has local maximum points at and and a local minimum point at .
- B has local maximum points at and and local minimum points at and .
- C has a local maximum point at and local minimum points at and .
- D has local maximum points at and and a local minimum point at .
- E has local maximum points at and and local minimum points at and .

**Q7: **

Given that the function has a minimum value of 2 at , determine the values of and .

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

**Q8: **

Find the local maximum and minimum values of .

- Alocal maximum at , local minimum 20 at
- Blocal maximum 20 at , local minimum at
- Clocal maximum 13 at , local minimum 4 at
- Dlocal maximum 8 at , local minimum 17 at
- Elocal maximum 4 at , local minimum 13 at

**Q9: **

Find the values of where has a local maximum or a local minimum.

- AThe function has a local maximum value at .
- BThe function has a local minimum value at .
- CThe function has a local minimum value at .
- DThe function has neither local maximum nor local minimum values.

**Q10: **

Determine (if there are any) the values of the local maximum and the local minimum of the function .

- ALocal maximum value, local minimum value
- BLocal maximum value
- Clocal minimum value
- DLocal minimum value
- ELocal minimum value, local maximum value