Critical numbers and the first derivative test

About this lesson

Calculus AB (AP-aligned) › Applications of derivatives

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Critical numbers and the first derivative test

Warm-up

Three quick questions from earlier lessons.

Discover

On a graph, a hilltop has a horizontal tangent. Just before it the graph rises; just after it falls. What does f′f' do at a hilltop?

  1. It goes from ++ to −-
  2. It goes from −- to ++

Key idea

Extremes happen where f′f' changes sign.

A critical number of ff is a cc where f′(c)=0f'(c) = 0 or f′(c)f'(c) does not exist. First derivative test: if f′f' changes from ++ to −- at cc, ff has a local maximum; from −- to ++, a local minimum; no sign change, no extremum.

Worked example

f(x)=x3−12xf(x) = x^3 - 12x. Find the local extremes.

f′(x)=3x2−12=3(x−2)(x+2)f'(x) = 3x^2 - 12 = 3(x - 2)(x + 2)
Factor f'.
Signs: + ∣ − ∣ ++\ |\ -\ |\ +
Sign chart: + before −2, − between, + after 2.
Local max at x=−2x = -2, local min at x=2x = 2
+ to − at −2; − to + at 2.

Your turn, guided

Your turn: f(x)=x2−6xf(x) = x^2 - 6x.

critical number x=3x = 3
f'(x) = 2x − 6 = 0.
local minimum at x=3x = 3
− before 3, + after: a minimum.

Practice ★

  1. f(x)=x3−12xf(x) = x^3 - 12x. At which xx does ff have a local maximum?
  2. f(x)=x3−27xf(x) = x^3 - 27x. At which xx does ff have a local maximum?
  1. Here is the graph of the derivative y=f′(x)y = f'(x). Tap the point of this graph at the xx-value where ff has a local maximum.
  2. Here is the graph of the derivative y=f′(x)y = f'(x). Tap the point of this graph at the xx-value where ff has a local maximum.

Watch out!

Practice ★★

  1. f′(x)=(x−−3)(x−5)2f'(x) = (x - -3)(x - 5)^2. Tick all the true statements.
    1. ff has a local minimum at x=−3x = -3
    2. ff is increasing for x>−3x > -3
    3. ff has a local maximum at x=−3x = -3
    4. ff has a local extremum at x=5x = 5
  2. f′(x)=(x−0)(x−4)2f'(x) = (x - 0)(x - 4)^2. Tick all the true statements.
    1. ff has a local minimum at x=0x = 0
    2. ff is increasing for x>0x > 0
    3. ff has a local maximum at x=0x = 0
    4. ff has a local extremum at x=4x = 4

Exit ticket

Three questions to check you have got it.

  1. f(x)=x3−12xf(x) = x^3 - 12x. At which xx does ff have a local maximum?
  2. f(x)=x3−27xf(x) = x^3 - 27x. At which xx does ff have a local maximum?
  3. f(x)=x3−3xf(x) = x^3 - 3x. At which xx does ff have a local maximum?

Summary card

First derivative test

  1. Find the critical numbers.
  2. Sign chart of f′f'.
  3. +→−+ \to -: max; −→+- \to +: min; no change: neither.

Watch out: f′(c)=0f'(c) = 0 alone does not make an extremum.

End of the lesson

Next lesson: The second derivative test and inflection points Back to the chapter

Answers: Critical numbers and the first derivative test

A

  1. −2-2
  2. −3-3

B

  1. (−3,0)(-3, 0)
  2. (−1,0)(-1, 0)

C

  1. ff has a local minimum at x=−3x = -3 · ff is increasing for x>−3x > -3
  2. ff has a local minimum at x=0x = 0 · ff is increasing for x>0x > 0

D

  1. −2-2
  2. −3-3
  3. −1-1

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