# 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