Question

Which of the following is NOT CORRECT?

• A. A quasiconcave function is necessarily a concave function
• B. A concave function is necessarily a quasiconcave function
• C. A quasiconcave function can also be a quasiconvex function
• D. A quasiconcave function can also be a convex function

Answer 1

The Correct Option is A

To determine the correct answer, let's explore the definitions and properties of quasiconcave, concave, quasiconvex, and convex functions. This requires a theoretical understanding of these mathematical functions: 

• Quasiconcave Function: A function \(f(x)\) is quasiconcave if for any two points \(x_1\) and \(x_2\) in its domain, and for any \(\lambda \in [0, 1]\), we have: \(f(\lambda x_1 + (1-\lambda)x_2) \geq \min\{f(x_1), f(x_2)\}\). Essentially, this means the upper contour set of the function is convex.
• Concave Function: A function \(f(x)\) is concave if for any two points \(x_1\) and \(x_2\) in its domain, and for any \(\lambda \in [0, 1]\), we have: \(f(\lambda x_1 + (1-\lambda)x_2) \geq \lambda f(x_1) + (1-\lambda)f(x_2)\). This implies that any line segment joining two points on the graph of the function lies below or on the graph.
• Quasiconvex Function: A function is quasiconvex if for any two points \(x_1\) and \(x_2\), and for any \(\lambda \in [0, 1]\), \(f(\lambda x_1 + (1-\lambda)x_2) \leq \max\{f(x_1), f(x_2)\}\). This suggests the lower contour set of the function is convex.
• Convex Function: A function \(f(x)\) is convex if for any two points \(x_1\) and \(x_2\), and for any \(\lambda \in [0, 1]\), \(f(\lambda x_1 + (1-\lambda)x_2) \leq \lambda f(x_1) + (1-\lambda)f(x_2)\). This means the line segment between any two points on its graph lies above or on the graph.

Now, addressing each option:

1. A quasiconcave function is necessarily a concave function: This is FALSE. A quasiconcave function is not necessarily concave. While all concave functions are quasiconcave (they satisfy the quasiconcavity condition), the converse is not true. Quasiconcave functions are more general and do not require the linear combination condition of concave functions.
2. A concave function is necessarily a quasiconcave function: This is TRUE. As mentioned, all concave functions meet the criteria of quasiconcavity.
3. A quasiconcave function can also be a quasiconvex function: This is TRUE. A function can be both quasiconcave and quasiconvex if it is a constant function.
4. A quasiconcave function can also be a convex function: This is TRUE. Like concavity, convexity is a stronger condition, and some quasiconcave functions can meet this condition.

Thus, the statement that is NOT correct is: A quasiconcave function is necessarily a concave function.