Answer: f'(x) = 0 or f'(x) does not exist.
- A f(x) = 0, meaning the function itself vanishes
- B f'(x) = 0 or f'(x) does not exist
- C f''(x) = 0 always, marking a guaranteed inflection point
- D f is discontinuous at that particular x-value
Correct answer: B. f'(x) = 0 or f'(x) does not exist
Explanation: Critical points occur where the derivative is zero or undefined; these are candidates for local maxima or minima.
The tangent at a point touches the curve with slope f'(a); the normal is the line perpendicular to the tangent at that same point, with slope -1/f'(a) - together they describe the curve's local direction and the line "straight into" the curve.
Concept context
Use derivatives to study rate of change, increasing and decreasing functions, tangents and normals, and maxima and minima, with classic optimization problems.