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

Start Practising

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)=−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.

Q4:

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

Q5:

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.

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=−𝑥−1𝑥+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

Practice Means Progress

Boost your grades with free daily practice questions. Download Nagwa Practice today!

scan me!

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