The portal has been deactivated. Please contact your portal admin.

Lesson Worksheet: Second Derivative Test for Local Extrema Mathematics • Higher Education

In this worksheet, we will practice classifying local extrema using the second derivative test.

Q1:

Find, if any, the points (π‘₯,𝑦) where 𝑦=βˆ’π‘₯+4π‘₯βˆ’6 has a local maximum or local minimum.

  • A(βˆ’2,βˆ’18)is a local maximum point.
  • B(βˆ’2,βˆ’18)is a local minimum point.
  • C(2,βˆ’2)is a local minimum point.
  • DThe function does not have local maximum or minimum points.
  • E(2,βˆ’2)is a local maximum point.

Q2:

Find, if any, the points (π‘₯,𝑦) where 𝑦=π‘₯+3π‘₯βˆ’16 has a local maximum or local minimum.

  • A(βˆ’2,βˆ’12)is the local minimum point, and the function does not have a local maximum point.
  • B(βˆ’2,βˆ’12)is the local maximum point, and (0,βˆ’16)is the local minimum point.
  • C(0,βˆ’16)is the local minimum point, and the function does not have a local maximum point.
  • D(βˆ’2,βˆ’12)is the local maximum point, and the function does not have a local minimum point.
  • E(βˆ’2,βˆ’12)is the local minimum point, and (0,βˆ’16)is the local maximum point.

Q3:

Find, if any, the local maximum and local minimum values of 𝑦=7π‘₯+7π‘₯.

  • AThe function does not have local maximum or minimum points.
  • BThe local maximum value isβˆ’14.
  • CThe local maximum value is14, and the local minimum value isβˆ’14.
  • DThe local minimum value is14, and the local maximum value isβˆ’14.
  • EThe local minimum value is14.

Q4:

Find the coordinates of all the local minima and maxima of the function 𝑓(π‘₯)=3π‘₯+5+6π‘₯.

  • AThere are no local minima or maxima values.
  • BLocal minima at ο€Ώβˆš22,5+6√2 and local maxima at ο€Ώβˆ’βˆš22,5βˆ’6√2
  • CLocal minima at ο€Ώβˆš22,5+6√2
  • DLocal minima at ο€Ώβˆ’βˆš22,5βˆ’6√2
  • ELocal minima at ο€Ώβˆ’βˆš22,5βˆ’6√2 and local maxima at ο€Ώβˆš22,5+6√2

Q5:

Find the local maxima/minima of the function 𝑓(π‘₯)=3π‘₯βˆ’2π‘₯οŠͺ.

  • Aο€Όβˆ’12,716 is a local maximum point.
  • Bο€Ό12,βˆ’116 is a local minimum point.
  • Cο€Όβˆ’12,716 is a local minimum point.
  • Dο€Ό12,βˆ’116 is a local maximum point.
  • EThe function does not have local maximum or minimum points.

Q6:

Find the local maximum and minimum values of 𝑓(π‘₯)=2π‘₯βˆ’π‘₯, if there are any. Round your answer to the nearest three decimal places.

  • AThe local minimum value =0.066 and the local maximum value =βˆ’0.066.
  • BThe local minimum values are 0.066 and βˆ’0.066.
  • CThe local maximum values are 0,0.066, and βˆ’0.066.
  • DThe local maximum values are 0.066 and βˆ’0.066.
  • EThe local minimum value =βˆ’0.066 and the local maximum value =0.066.

Q7:

Find the local maximum and minimum values of 𝑓(π‘₯)=π‘’βˆ’π‘’οŠ¨ο—, if any.

  • AThe local maximum value is 𝑒.
  • BThe local minimum value is 𝑒.
  • CThe local minimum value is 0.
  • DThe local maximum value is 0.
  • E𝑓 does not have any local maximum or minimum values.

Q8:

Find the local maximum and minimum values of β„Ž(π‘₯)=2+(π‘₯βˆ’1), if any.

  • AThe local maximum value is 2 and the local minimum value is βˆ’6.
  • BThe local maximum value is 0.
  • CThe local minimum value is 2.
  • DThe local maximum value is 2.
  • Eβ„Ž does not have any local maximum or minimum values.

Q9:

Find the local maximum and minimum values of 𝑓(π‘₯)=2√π‘₯βˆ’4√π‘₯.

  • Alocal minimum 0 at π‘₯=16
  • Blocal maximum 0 at π‘₯=16
  • Clocal minimum βˆ’2 at π‘₯=1
  • Dno local maximum and no local minimum values
  • Elocal maximum βˆ’2 at π‘₯=1

Q10:

Find the local maxima and local minima of 𝑓(π‘₯)=βˆ’5π‘₯3+2π‘₯βˆ’16π‘₯ln, if any.

  • Alocal minimum 712βˆ’16ο€Ό12ln at π‘₯=12 , local maximum 1160βˆ’16ο€Ό110ln at π‘₯=110
  • Blocal minimum 815βˆ’16ο€Ό25ln at π‘₯=25 , local maximum βˆ’83βˆ’162ln at π‘₯=2
  • Clocal minimum 13βˆ’16ο€Ό15ln at π‘₯=15 , local maximum 13 at π‘₯=13
  • Dlocal minimum 13 at π‘₯=1 , local maximum 13βˆ’16ο€Ό15ln at π‘₯=15
  • Elocal minimum 1160βˆ’16ο€Ό110ln at π‘₯=110 , local maximum 712βˆ’16ο€Ό12ln at π‘₯=12

This lesson includes 23 additional questions and 226 additional question variations for subscribers.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.