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Lesson Worksheet: Critical Points and Local Extrema of a Function Mathematics • 12th Grade
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:
Determine the critical points of the function in the interval .
- A,
- B
- C,
- D,
- E
Q3:
Which graph has three real zeros and two local maxima?
- A
- B
- C
Q4:
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 .
Q5:
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
Q6:
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 .
Q7:
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 .
Q8:
Find, if any, the local maxima and minima for .
- ALocal maximum equals 1 at .
- BLocal minimum equals at .
- CIt has no local maxima and no local minima.
- DLocal minimum equals 1 at .
- ELocal maximum equals at .
Q9:
Given that the function has a minimum value of 2 at , determine the values of and .
- A,
- B,
- C,
- D,
Q10:
Determine where has a local maximum, and give the value there.
- A.
- B.
- C.
- D.
- E.