Which statement defines a critical point?

Which statement defines a critical point?

• A. They are always maxima.
• B. They are endpoints only.
• C. They are interior points where f' is negative.
• D. They are points where f'(x) = 0 or f'(x) is undefined.

Answer: (D)

A point is critical when the slope of the tangent is zero or the derivative fails to exist there, and it must lie in the function’s domain. That captures both horizontal tangents (where f′(x)=0) and nondifferentiable points like cusps or corners (where f′ is undefined). Think of examples: the graph of f(x)=x^3 has a horizontal tangent at x=0, and f′(0)=0, a critical point; the graph of f(x)=|x| has a cusp at x=0 where the derivative doesn’t exist, also a critical point. Saying that critical points are always maxima is too restrictive, since a critical point can correspond to a minima or just a point of inflection with a horizontal tangent. Endpoints aren’t the general focus of the definition because criticality concerns where the derivative is zero or undefined in the interior. And having a negative derivative simply means the function is decreasing there, not that the point is critical.

A point is critical when the slope of the tangent is zero or the derivative fails to exist there, and it must lie in the function’s domain. That captures both horizontal tangents (where f′(x)=0) and nondifferentiable points like cusps or corners (where f′ is undefined). Think of examples: the graph of f(x)=x^3 has a horizontal tangent at x=0, and f′(0)=0, a critical point; the graph of f(x)=|x| has a cusp at x=0 where the derivative doesn’t exist, also a critical point.

Saying that critical points are always maxima is too restrictive, since a critical point can correspond to a minima or just a point of inflection with a horizontal tangent. Endpoints aren’t the general focus of the definition because criticality concerns where the derivative is zero or undefined in the interior. And having a negative derivative simply means the function is decreasing there, not that the point is critical.