Worksheet: Local Extrema and Critical Points

Q1:

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

AThe function does not have local maximum or minimum points.
Bis a local maximum point.
Cis a local minimum point.
Dis a local minimum point.
Eis a local maximum point.

Q2:

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

AThe function does not have local maximum or minimum points.
Bis a local minimum point.
Cis a local maximum point.
Dis a local maximum point.
Eis a local minimum point.

Q3:

Find the local maximum and minimum values of .

Alocal maximum 4 at , local minimum 40 at
Blocal maximum at , local minimum at
Clocal maximum 28 at , local minimum at
Dlocal maximum at , local minimum 28 at
Elocal maximum at , local minimum at

Q4:

Determine the local maximum and local minimum values of .

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

Q5:

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 .

Q6:

Find the local maxima/minima of the function .

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

Q7:

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

Alocal minimum value
Blocal minimum value , local maximum value
Clocal maximum value
Dlocal 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 , local minima at and
Clocal maxima at , local minima at and
Dlocal maxima at and , local minima at

Q9:

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

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

Q10:

Find the local maximum and minimum values 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

Q11:

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

AIt has no local maximum or minimum values.
Blocal minimum value, local maximum value
Clocal minimum value, local maximum value
Dlocal minimum value, local maximum value
Elocal minimum value, local maximum value

Q12:

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

AThe function doesn’t have local maximum or minimum points.
Bis a local maximum point.
Cis a local minimum point, and is a local maximum point.
Dis a local minimum point.
Eis a local maximum point, and is a local minimum point.

Q13:

Find the local maximum and minimum values of .

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

Q14:

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

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

Q15:

Determine the local maximum and minimum values of the function .

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

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

Q18:

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

Alocal maximum at .
Blocal minimum at .
Clocal maximum at .
Dlocal maximum at .
Elocal minimum 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 .

Ano local maximum and no local minimum values
Blocal maximum 0 at , local minimum at
Clocal maximum at , local minimum 0 at
Dlocal maximum at , local minimum 0 at
Elocal maximum 0 at , local minimum at

Q21:

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

Alocal maximum value , local minimum value
BThe function has no local minimum or local maximum values.
Clocal minimum value , local maximum 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 minimum
D, local minimum
E, local maximum