Question

If a continuous function $f$ has a relative extremum at $c$ , then $c$ must be

• A. a cusp of $f$
• B. a critical number of $f$
• C. a pole of $f$
• D. a horizontal asymptote of $f$

Hint:

Every local extremum occurs at a critical point, but not every critical point is a local extremum (e.g., $f(x) = x^3$ has a critical point at $x=0$, but it is an inflection point, not an extremum).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Fermat's Theorem on stationary points describes the relationship between local extrema and the derivative of a function.

Step 2: Detailed Explanation:
A critical number of a continuous function $f$ is defined as a value $c$ in the domain of $f$ where either: \[ f'(c) = 0 \] or $f'(c)$ is undefined (does not exist).
According to Fermat's Theorem, if a function $f$ has a local (relative) maximum or local minimum at $x = c$, and if $f$ is differentiable at $c$, then $f'(c) = 0$.
If the function is not differentiable at the extremum point (such as at a sharp corner or cusp), then $f'(c)$ does not exist, which still fits the definition of a critical number.
Therefore, any point where a relative extremum occurs must be a critical number of the function.
A pole is a concept from complex analysis where a function goes to infinity, and a horizontal asymptote describes the limit of a function as $x \to \pm\infty$.

Step 3: Final Answer:
The value $c$ must be a critical number of $f$.