# Worksheet: Critical Points and Local Extrema of a Function

In this worksheet, we will practice finding critical points of a function and checking for local extrema using the first derivative test.

Q1:

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

Q2:

Determine the critical points of the function in the interval .

• A,
• B
• C,
• D,
• E

Q3:

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.

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 (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

Q6:

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

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

Q7:

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

• AThe function has no local minimum or local maximum values.
• Blocal minimum value , local maximum value
• Clocal maximum value , local minimum value

Q8:

Given that the function has a minimum value of 2 at , determine the values of and .

• A,
• B,
• C,
• D,

Q9:

Determine the critical points of the function in the interval .

• A, , , ,
• B, , , ,
• C, ,
• D, ,

Q10:

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 .

Q11:

Find, if any, the points 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, and is a local minimum point.
• D is a local minimum point.
• E is a local minimum point, and is a local maximum point.

Q12:

Find the local maxima and minima of , if any.

• Alocal minimum value is at
• Blocal maximum value is at
• Clocal maximum value is at
• Dlocal minimum value is at
• Elocal minimum value is at

Q13:

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 .

Q14:

Find, if any, the local maximum and local minimum values of , together with their type. Give your answers to two decimal places.

• AThe function has no local maximum or minimum points.
• B, local minimum value
• C, local minimum value
• D, local maximum value
• E, local maximum value

Q15:

The figure shows the graph of for . Give an exact expression for the -coordinate of the point , including if necessary.

• A
• B
• C
• D
• E

Q16:

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

• AThe function has no local maxima or minima.
• Blocal maximum at
• Clocal minimum at
• Dlocal maximum at
• Elocal minimum at

Q17:

Determine the value of where the function has a critical point.

Q18:

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

• AThere are no local minima or maxima.
• Blocal maximum at
• Clocal maximum 3 at
• Dlocal minimum at
• Elocal minimum 3 at

Q19:

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

• Alocal maximum at
• Blocal maximum at
• Clocal minimum at
• Dlocal minimum 1 at
• Elocal maximum 1 at

Q20:

The function has a critical number at . Find and list all the critical numbers.

• A, , ,
• B, , ,
• C, , ,
• D, , ,
• E, , ,

Q21:

Determine where has a local maximum, and give the value there.

• A.
• B.
• C.
• D.
• E.

Q22:

Determine the local maximum and minimum values of the function .

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

Q23:

Determine, if any, the local maximum/minimum values for the function .

• Alocal maximum value:
• Blocal maximum value:
• Clocal minimum value:
• Dlocal minimum value:
• EThere are no local maxima/minima.

Q24:

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

• Alocal maximum at
• Blocal minimum at
• Clocal minimum at
• Dlocal maximum at
• Ehas no local maxima and no local minima

Q25:

Determine the critical points of the function in the interval .

• A,
• B, ,
• C, ,
• D,
• E,