Worksheet: Local Extrema and Critical Points

Q1:

Find, if any, the point where has a local maximum or local minimum.

A is a local minimum point.
B is a local maximum point.
C is a local maximum point.
D is a local minimum point.
EThe function does not have local maximum or minimum points.

Q2:

Find, if any, the point where has a local maximum or local minimum.

A is a local maximum point.
B is a local minimum point.
C is a local minimum point.
D is a local maximum point.
EThe function does not have local maximum or minimum points.

Q3:

Find the local maximum and minimum values of .

A local maximum at , local minimum 28 at
B local maximum at , local minimum at
C local maximum at , local minimum at
D local maximum 28 at , local minimum at
E local maximum 4 at , local minimum 40 at

Q4:

Determine the local maximum and local minimum values of .

Alocal maximum value at , local minimum value 3 at
Blocal maximum value at , local minimum value at
Clocal maximum value 8 at , local minimum value 8 at
Dlocal maximum value 3 at , local minimum value at
Elocal maximum value 8 at , local minimum value 8 at

Q5:

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

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

Q6:

Find the local maxima/minima of the function .

A is a local minimum point.
B is a local maximum point.
C is a local maximum point.
D is a local minimum point.
E The function does not have local maximum or minimum points.

Q7:

Use the second derivative test to find, if possible, the local maximum and minimum values of .

A local maximum value
B local minimum value , local maximum value
C local minimum value
D local minimum value , local maximum value

Q8:

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 , local minima at and

Q9:

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

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

Q10:

Find the local maximum and minimum values of .

Alocal maximum at , local minimum at
B local maximum at , local minimum at
C local maximum at , local minimum at
D local maximum at , local minimum at
E local maximum at , local minimum at

Q11:

Find where, if at all, the function has its local maxima and minima.

Alocal minimum value, local maximum value
Blocal minimum value, local maximum value
Clocal minimum value, local maximum value
Dlocal minimum value, local maximum value
E It has no local maximum or minimum values.

Q12:

Find the points where has a local maximum or a local minimum.

A is a local minimum point.
B is a local maximum point, and is a local minimum point.
C is a local maximum point.
D is a local minimum point, and is a local maximum point.
EThe function doesn’t have local maximum or minimum points.

Q13:

Find the local maximum and minimum values of .

AA local maximum value is at .
BA local minimum value is at .
CA local minimum value is at .
DA local maximum value is at .
EA local maximum value is at .

Q14:

Find where (if at all) the function has its local maxima and minima.

Alocal maximum value
Blocal minimum value, local maximum value
Clocal minimum value
Dlocal maximum value, local minimum value
Elocal minimum value

Q15:

Determine the local maximum and minimum values of the function .

Alocal minimum , local maximum
Blocal maximum , local minimum
Clocal maximum , local minimum
Dlocal minimum , local maximum

Q16:

Locate and classify the critical points of .

Alocal maximum at , local minimum at
Blocal maximum at , local minimum at
Clocal maximum at , local minimum at
Dlocal maximum at , local minimum at
Elocal maximum at , local minimum at

Q17:

Find the local maximum and local minimum values of the function , if these exist.

Alocal minimum value , no local maximum value
Blocal minimum value , local maximum value
Clocal maximum value , no local minimum value
Dlocal maximum value , local minimum value

Q18:

Find (if any) the local maxima and local minima of .

A local maximum at .
B local minimum at .
C local minimum at .
D local maximum at .
E local maximum at .

Q19:

Let , find where the local maxima and local minima occur for .

Alocal maximum at , local minimum at
Blocal maximum at , local minimum at
Clocal maximum at , local minimum at
Dlocal maximum at , local minimum at
Elocal maximum at , local minimum at

Q20:

Find the local maximum and minimum values of .

Alocal maximum at , local minimum 0 at
Blocal maximum 0 at , local minimum at
Clocal maximum 0 at , local minimum at
Dlocal maximum at , local minimum 0 at
Eno local maximum and no local minimum values

Q21:

Determine the local minimum and local maximum values of the function .

AThe function has no local minimum or local maximum values.
B local minimum value , local maximum value
C local maximum value , local minimum value

Q22:

Find the point where has a critical point, and determine whether it is a local maximum or local minimum.

A , local minimum
B , local maximum
C , local maximum
D , local minimum
E , local minimum