Question

The point of inflexion of a function \( f(x) \) is the point where: {5pt}

• A. \( f'(x) = 0 \) and \( f'(x) \) changes its sign from positive to negative from left to right of that point.
• B. \( f'(x) = 0 \) and \( f'(x) \) changes its sign from negative to positive from left to right of that point.
• C. \( f'(x) = 0 \) and \( f'(x) \) does not change its sign from left to right of that point.
• D. \( f'(x) \neq 0 \).
  {5pt}

Hint:

For points of inflection: - Check \( f''(x) \) for sign changes. - \( f'(x) \) does not change sign at a point of inflection.

Answer 1

The Correct Option is C

Step 1: Definition of an inflection point. A point of inflection is where a function's concavity transitions, indicated by a sign change in \( f''(x) \).
Step 2: Criteria for inflection points. For a function \( f(x) \) to have an inflection point at \( x = c \), two conditions must be met: \( f''(x) \) must change sign around \( x = c \), and if \( f'(c) = 0 \), \( f'(x) \) must not change sign at \( c \). 
Step 3: Evaluation of choices. Option (C) accurately describes the condition \( f'(x) = 0 \) without a sign change in \( f'(x) \), aligning with the properties of an inflection point. 
Step 4: Determination. Option (C) is the correct answer.