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