Question

If f is continuous on an open interval containing a value $x_0$, and if f changes the direction of its concavity at the point ($x_0, f(x_0)$), then f has :

• A. a relative maximum at $x_0$
• B. a relative minimum at $x_0$
• C. an inflection point at $x_0$
• D. a saddle point at $x_0$

Hint:

An inflection point occurs where the second derivative $f''(x)$ changes sign.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Concavity refers to the direction of curvature of a graph.
A graph is concave up if it curves upwards (like a cup) and concave down if it curves downwards (like a cap).
Detailed Explanation:
Let us analyze the definitions of the options:
- Relative maximum: A point where the function changes from increasing to decreasing. The concavity is usually down, but does not necessarily change.
- Relative minimum: A point where the function changes from decreasing to increasing.
- Inflection point: By definition, a point on a curve $(x_0, f(x_0))$ where the direction of concavity changes (either from concave up to concave down, or from concave down to concave up) is called an inflection point.
- Saddle point: A point on a surface (functions of two variables) where the slopes in orthogonal directions are zero but it is not a local extremum.
Since the question specifies a change in the direction of concavity, the point must be an inflection point.

Step 2: Final Answer:
The point is an inflection point, which corresponds to Option (C).