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Which of the following statements is FALSE for a differentiable function f(x)?

  1. At a critical point of f(x), the derivative of f(x) is either 0 or undefined.
  2. A critical point of f(x) must belong to the domain of f(x).
  3. f(x) has at least one critical point.
  4. A local extremum of f(x) must occur at a critical point of f(x).

Answer (Detailed Solution Below)

Option 3 : f(x) has at least one critical point.
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Detailed Solution

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Explanation:

  • Statements A and B are true by definition of a critical point. Statement D is also true.
  • Statement C is false: f(x) = 1 / x has no critical points. The point x = 0 is not considered.as a critical point since it is not included in the domain of \(f\left( x \right)\) 

i.e. \(f'\left( x \right) = - 1/{x^2}\) is neither 0 nor undefined, for all x in the domain of f(x) as the domain of f(x) = 1/x is R - {0}.

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