CalcRef

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.

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.

Find the critical points of f(x) = x^(2/3).
  1. f'(x) = (2/3)x^(-1/3) = 2/(3x^(1/3))
  2. f'(x) is undefined at x = 0 and never equals 0

Answer: Critical point at x = 0 (derivative undefined).

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