# 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.

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