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Lesson Worksheet: Critical Points and Local Extrema of a Function Mathematics • 12th Grade

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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 𝑦=−8𝑥 in the interval [−2,1].

  • A(0,0), (1,−8)
  • B(0,0)
  • C(0,0), (1,−24)
  • D(−2,64), (1,−8)
  • E(−2,64)

Q3:

Which graph has three real zeros and two local maxima?

  • A()b
  • B()a
  • C()c

Q4:

Determine, if any, the local maximum and minimum values of 𝑓(𝑥)=−2𝑥−9𝑥−12𝑥−15, together with where they occur.

  • AThe local maximum is −38 at 𝑥=1, and there is no local minimum.
  • BThe local maximum is −10 at 𝑥=−1, and the local minimum is −11 at 𝑥=−2.
  • CThe local minimum is −15 at 𝑥=0, and there is no local maximum.
  • DThe local minimum is −2 at 𝑥=−14, and the local maximum is −15 at 𝑥=29.

Q5:

Determine where the local maxima and minima are for 𝑓(𝑥)=𝑥4−2𝑥+5.

  • Alocal maxima at 𝑥=0, local minima at 𝑥=4 and 𝑥=−4
  • Blocal maxima at 𝑥=2 and 𝑥=−2, local minima at 𝑥=0
  • Clocal maxima at 𝑥=0, local minima at 𝑥=2 and 𝑥=−2
  • Dlocal maxima at 𝑥=4 and 𝑥=−4, local minima at 𝑥=0
  • Elocal maxima at 𝑥=2 and 𝑥=0, local minima at 𝑥=−2

Q6:

Find the critical points of 𝑓(𝑥)=𝑥(𝑥−1).

  • AThe function has critical points at 𝑥=0, 𝑥=−25, and 𝑥=−1.
  • BThe function has critical points at 𝑥=0, 𝑥=1, and 𝑥=13.
  • CThe function has no critical points.
  • DThe function has critical points at 𝑥=0, 𝑥=25, and 𝑥=1.
  • EThe function has critical points at 𝑥=0, 𝑥=−1, and 𝑥=−13.

Q7:

The graph of the first derivative 𝑓′ of a continuous function 𝑓 is shown. At what values of 𝑥 does 𝑓 have a local maximum and a local minimum?

  • A𝑓 has local maximum points at 𝑥=1 and 𝑥=8 and a local minimum point at 𝑥=6.
  • B𝑓 has local maximum points at 𝑥=2 and 𝑥=5 and local minimum points at 𝑥=3 and 𝑥=7.
  • C𝑓 has a local maximum point at 𝑥=6 and local minimum points at 𝑥=1 and 𝑥=8.
  • D𝑓 has local maximum points at 𝑥=3 and 𝑥=5 and a local minimum point at 𝑥=7.
  • E𝑓 has local maximum points at 𝑥=3 and 𝑥=7 and local minimum points at 𝑥=2 and 𝑥=5.

Q8:

Find, if any, the local maxima and minima for 𝑓(𝑥)=3𝑥−2𝑥−4𝑥ln.

  • ALocal maximum equals 1 at 𝑥=1.
  • BLocal minimum equals −423ln at 𝑥=23.
  • CIt has no local maxima and no local minima.
  • DLocal minimum equals 1 at 𝑥=1.
  • ELocal maximum equals −423ln at 𝑥=23.

Q9:

Given that the function 𝑓(𝑥)=𝑥+𝐿𝑥+𝑀 has a minimum value of 2 at 𝑥=−1, determine the values of 𝐿 and 𝑀.

  • A𝐿=−4, 𝑀=−3
  • B𝐿=2, 𝑀=3
  • C𝐿=1, 𝑀=2
  • D𝐿=−2, 𝑀=−1

Q10:

Determine where 𝑓(𝑥)=3𝑥𝑒 has a local maximum, and give the value there.

  • A𝑥=12,34√𝑒.
  • B𝑥=−2,12𝑒.
  • C𝑥=23,43𝑒.
  • D𝑥=−12,3√𝑒4.
  • E𝑥=2,12𝑒.

This lesson includes 82 additional questions and 415 additional question variations for subscribers.

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