Lesson Worksheet: Critical Points and Local Extrema of a Function Mathematics

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:

Which graph has three real zeros and two local maxima?

  • A()b
  • B()a
  • C()c

Q3:

Determine, if any, the local maximum and minimum values of 𝑓(𝑥)=2𝑥9𝑥12𝑥15, 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.

Q4:

Determine where the local maxima and minima are for 𝑓(𝑥)=𝑥42𝑥+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

Q5:

Find the critical points of 𝑓(𝑥)=𝑥(𝑥1).

  • 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.

Q6:

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.

Q7:

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

Q8:

Find the local maximum and minimum values of 𝑓(𝑥)=2𝑥+3𝑥+12𝑥.

  • Alocal maximum 7 at 𝑥=1, local minimum 20 at 𝑥=2
  • Blocal maximum 20 at 𝑥=2, local minimum 7 at 𝑥=1
  • Clocal maximum 13 at 𝑥=1, local minimum 4 at 𝑥=2
  • Dlocal maximum 8 at 𝑥=2, local minimum 17 at 𝑥=20
  • Elocal maximum 4 at 𝑥=2, local minimum 13 at 𝑥=1

Q9:

Find the values of 𝑥 where 𝑓(𝑥)=(𝑥+4) has a local maximum or a local minimum.

  • AThe function has a local maximum value at 𝑥=4.
  • BThe function has a local minimum value at 𝑥=4.
  • CThe function has a local minimum value at 𝑥=4.
  • DThe function has neither local maximum nor local minimum values.

Q10:

Determine (if there are any) the values of the local maximum and the local minimum of the function 𝑦=𝑥1𝑥+8.

  • ALocal maximum value=6, local minimum value=10
  • BLocal maximum value=6
  • Clocal minimum value=10
  • DLocal minimum value=6
  • ELocal minimum value=6, local maximum value=10

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