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

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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=βˆ’8π‘₯ in the interval [βˆ’2,1].

  • A(0,0), (1,βˆ’8)
  • B(0,0)
  • C(0,0), (1,βˆ’24)
  • D(βˆ’2,64), (1,βˆ’8)
  • E(βˆ’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(π‘₯)=βˆ’2π‘₯βˆ’9π‘₯βˆ’12π‘₯βˆ’15, together with where they occur.

  • AThe local maximum is βˆ’38 at π‘₯=1, and there is no local minimum.
  • BThe local maximum is βˆ’10 at π‘₯=βˆ’1, and the local minimum is βˆ’11 at π‘₯=βˆ’2.
  • CThe local minimum is βˆ’15 at π‘₯=0, and there is no local maximum.
  • DThe local minimum is βˆ’2 at π‘₯=βˆ’14, and the local maximum is βˆ’15 at π‘₯=29.

Q5:

Determine where the local maxima and minima are for 𝑓(π‘₯)=π‘₯4βˆ’2π‘₯+5οŠͺ.

  • Alocal maxima at π‘₯=0, local minima at π‘₯=4 and π‘₯=βˆ’4
  • Blocal maxima at π‘₯=2 and π‘₯=βˆ’2, local minima at π‘₯=0
  • Clocal maxima at π‘₯=0, local minima at π‘₯=2 and π‘₯=βˆ’2
  • Dlocal maxima at π‘₯=4 and π‘₯=βˆ’4, local minima at π‘₯=0
  • Elocal maxima at π‘₯=2 and π‘₯=0, local minima at π‘₯=βˆ’2

Q6:

Find the critical points of 𝑓(π‘₯)=π‘₯(π‘₯βˆ’1).

  • AThe function has critical points at π‘₯=0, π‘₯=βˆ’25, and π‘₯=βˆ’1.
  • BThe function has critical points at π‘₯=0, π‘₯=1, and π‘₯=13.
  • CThe function has no critical points.
  • DThe function has critical points at π‘₯=0, π‘₯=25, and π‘₯=1.
  • EThe function has critical points at π‘₯=0, π‘₯=βˆ’1, and π‘₯=βˆ’13.

Q7:

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

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

Q8:

Find, if any, the local maxima and minima for 𝑓(π‘₯)=3π‘₯βˆ’2π‘₯βˆ’4π‘₯ln.

  • ALocal maximum equals 1 at π‘₯=1.
  • BLocal minimum equals βˆ’4ο€Ό23ln at π‘₯=23.
  • CIt has no local maxima and no local minima.
  • DLocal minimum equals 1 at π‘₯=1.
  • ELocal maximum equals βˆ’4ο€Ό23ln at π‘₯=23.

Q9:

Given that the function 𝑓(π‘₯)=π‘₯+𝐿π‘₯+π‘€οŠ¨ has a minimum value of 2 at π‘₯=βˆ’1, determine the values of 𝐿 and 𝑀.

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

Q10:

Determine where 𝑓(π‘₯)=3π‘₯π‘’οŠ¨οŠ±ο— has a local maximum, and give the value there.

  • Aπ‘₯=12,34βˆšπ‘’.
  • Bπ‘₯=βˆ’2,12π‘’οŠ¨.
  • Cπ‘₯=23,43π‘’οŽ‘οŽ’.
  • Dπ‘₯=βˆ’12,3βˆšπ‘’4.
  • Eπ‘₯=2,12π‘’οŠ¨.

This lesson includes 54 additional questions and 406 additional question variations for subscribers.

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