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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