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Applications of Derivatives · Curve Analysis

Critical Points

f′(c)=0 or f′(c) DNE⇒c is a critical pointf'(c) = 0 \text{ or } f'(c) \text{ DNE} \Rightarrow c \text{ is a critical point}

Critical points occur where the derivative is zero or undefined. These are candidates for local extrema.

  • 2 worked examples
  • 15 in Applications of Derivatives

Worked examples

Find the critical points of f(x) = x³ - 3x + 1.
  1. f'(x) = 3x² - 3 = 3(x² - 1) = 3(x-1)(x+1)
  2. f'(x) = 0 when x = 1 and x = -1

Answer: Critical points at x = -1 and x = 1.

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Find the critical points of f(x) = x^(2/3).

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Common questions

What is the Critical Points?

Critical points occur where the derivative is zero or undefined. These are candidates for local extrema. It is one of the applications of derivatives formulas in the CalcRef reference.

How do you use the Critical Points?

Worked example. Find the critical points of f(x) = x³ - 3x + 1. f'(x) = 3x² - 3 = 3(x² - 1) = 3(x-1)(x+1). f'(x) = 0 when x = 1 and x = -1. Answer: Critical points at x = -1 and x = 1.

Practice

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