Concave and quasi-concave functions Show that if f(x1, x2) is a concave function

Question

Concave and quasi-concave functions Show that if f(x1, x2) 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.

Explanation / Answer

Quasi concave functions have the property that for any two points in the domain, say x1 and x2, the value of f(x) on all points between them satisfies: f(x) >= min{f(x1), f(x2)} and this is true is if has a positive slope for all x in the interval which in your case is given by dc/dy= y which is >0 for all y>0 A concave function is a continuous function whose value at the midpoint of every interval in its domain is less than or equal to the arithmetic mean of its values at the ends of the interval. That is f( {x1+x2}/2 )

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