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 quasiconvex and its converse also holds.

Hint:

A function being convex does not necessarily imply it is quasiconvex, but every convex function is quasiconvex. Remember that quasiconvexity does not require strict convexity.

Answer 1

The Correct Option is D

- Quasiconcave: A function f(x) is quasiconcave if for any two points x₁ and x₂ in the domain, the line segment joining f(x₁) and f(x₂) lies entirely above the graph of the function.
- Quasiconvex: A function f(x) is quasiconvex if for any two points x₁ and x₂, the line segment joining f(x₁) and f(x₂) lies entirely below the graph of the function. • (A) If f(x) is quasiconcave, then -f(x) is quasiconvex: Correct. If a function is quasiconcave, then its negative is quasiconvex. • (B) If f(x) is a linear function, then it is quasiconcave as well as quasiconvex: Correct. A linear function is both quasiconcave and quasiconvex because it satisfies both the conditions for convex and concave functions. • (C) Any concave function is quasiconcave, but the converse is not true: Correct. Every concave function is quasiconcave, but not all quasiconcave functions are concave. • (D) Any convex function is quasiconvex and its converse also holds: Incorrect. While any convex function is quasiconvex, the converse is not true. There are quasiconvex functions that are not convex, making this statement false. The correct answer is (D) because the converse of the statement is not true: not every quasiconvex function is convex. Final Answer: Any convex function is quasiconvex and its converse also holds.