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: Understanding 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. The relationships between these concepts can help identify the incorrect statement.

Step 2: Analysis of options.
- (A) If \( f(x) \) is quasiconcave, then \( -f(x) \) is quasiconvex. This is correct because the negation of a quasiconcave function is quasiconvex.
- (B) If \( f(x) \) is a linear function, then it is quasiconcave as well as quasiconvex. This is correct because linear functions are both quasiconcave and quasiconvex.
- (C) Any concave function is quasiconcave, but the converse is not true. This is correct. A concave function is always quasiconcave, but not every quasiconcave function is concave.
- (D) Any convex function is quasiconcave, and its converse also holds. This is incorrect because not every convex function is quasiconcave. For instance, a convex function that is not concave may not be quasiconcave.

Step 3: Conclusion.
The incorrect statement is (D), which claims that the converse of convexity implies quasiconcavity, which is not true.