Show that if \(f\left(x_{1}, x_{2}\right)\) is a concave function then it is also a quasi-concave function. Do this by comparing Equation 2.114 (defining quasi-concavity) with Equation 2.98 (defining concavity). Can you give an intuitive reason for this result? Is the converse of the statement true? Are quasi-concave functions necessarily concave? If not, give a counterexample.

Question: Prove that a concave function is also a quasi-concave function. Answer: To prove that a concave function is also a quasi-concave function, we need to show that if function \(f\) is concave, it also satisfies the definition of a quasi-concave function. Since, for any two points \(x\) and \(y\) and for any scalar \(\alpha \in [0,1]\), a concave function \(f\) satisfies: $$ f(\alpha x + (1-\alpha) y) \ge \alpha f(x) + (1-\alpha) f(y) $$ We can see that for any \(x, y\) and \(\alpha \in [0,1]\), if \(f(x) \le f(y)\), then $$ f(\alpha x + (1-\alpha) y) \ge f(x) $$ and if \(f(x) > f(y)\), then $$ f(\alpha x + (1-\alpha) y) \ge f(y) $$ In both cases, we have that $$ f(\alpha x + (1-\alpha) y) \ge \min\{f(x), f(y)\} $$ meaning that \(f\) is also quasi-concave.