Question

Which of the following statements is not correct?

• A. If \( f(x) \) is quasiconcave then \( -f(x) \) is quasiconvex.
• B. If \( f(x) \) is a linear function, then it is quasiconcave as well as quasiconvex.
• C. Any concave function is quasiconcave but the converse is not true.
• D. Any convex function is quasiconcave and its converse also holds.

Hint:

Remember that convexity does not always imply quasiconcavity. A convex function may not be quasiconcave if it does not satisfy the upper-level set condition for quasiconcavity.

Answer 1

The Correct Option is D

Step 1: Define Quasiconcavity and Quasiconvexity. A function is quasiconcave if its upper level sets are convex. A function is quasiconvex if its lower level sets are convex. Understanding these definitions is crucial for identifying the incorrect statement.

Step 2: Evaluate Each Option.

• (A) If \( f(x) \) is quasiconcave, then \( -f(x) \) is quasiconvex. This statement is accurate; negating a quasiconcave function results in a quasiconvex function.
• (B) A linear function is both quasiconcave and quasiconvex. This statement is accurate; linear functions possess both properties.
• (C) All concave functions are quasiconcave, but the reverse is not necessarily true. This statement is accurate; while concavity implies quasiconcavity, the converse is not guaranteed.
• (D) All convex functions are quasiconcave, and vice versa. This statement is incorrect. A convex function is not always quasiconcave. For example, a convex function that is not concave may not be quasiconcave.

Step 3: Identify the Incorrect Statement. Statement (D) is incorrect because it falsely asserts that convexity implies quasiconcavity and that this relationship is bidirectional.