Topic: Properties of Quasiconvex and Quasiconcave Functions
Understanding the Question:
We need to identify the incorrect statement among the four options regarding the relationship between convexity/concavity and quasiconvexity/quasiconcavity.
Key Formulas and Approach:
A function is quasiconvex if its lower contour sets are convex.
All convex functions are quasiconvex, but not all quasiconvex functions are convex.
Detailed Solution:
Step 1: Verify negative transformations. If $f(x)$ is quasiconcave, its upper contour sets are convex. Multiplying by $-1$ flips the orientation, making the lower contour sets of $-f(x)$ convex. Thus, statement (A) is true.
Step 2: Check linear functions. Linear functions are both concave and convex. Since concavity implies quasiconcavity and convexity implies quasiconvexity, a linear function must be both. Thus, statement (B) is true.
Step 3: Evaluate the relationship between Concavity and Quasiconcavity. Every concave function is quasiconcave, but a monotonic function (like $f(x)=x^3$) is quasiconcave without being concave. Thus, statement (C) is true.
Step 4: Analyze the Converse of Convexity. While every convex function is quasiconvex, the converse (that every quasiconvex function is convex) is false. For example, $f(x) = \sqrt{|x|}$ is quasiconvex but not convex.
Conclusion: Statement (D) is incorrect because the converse does not hold.