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I would like to know if there is a theorem which links the quasi concavity of a function to the sign of its second order derivative. For eg. we know a function is Concave in a given interval if it's second order derivative is positive on a interval or that it convex if it's second order derivative is negative in a given interval . So on the same lines is there any test for quasi-concavity ?

Noob101
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2 Answers2

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I don't think it's possible to link quasiconcavity to the second derivative. As you note, concave functions have a negative 2nd derivative, and they are also quasiconcave. However, $e^{-|x|}$ (for example) is also quasiconcave but with positive 2nd derivative everywhere except zero (where it's undefined). You could also transform $e^{-|x|}$ to add arbitrary regions where its 2nd derivative is negative without changing its quasiconcavity, as long as the transformation preserves the monotonicity over $(-\infty,0)$ (increasing) and $(0,\infty)$ (decreasing) so that the only local maximum is at $x=0$.

David M Kaplan
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Quasi-concavity is a weaker condition than concavity, so it does not manifest in a condition on the sign of the second derivative. However, it is worth noting one useful characterisation that operates on the sign of the first derivative. If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a (strictly) quasi-concave continuously differentiable function, then it must have a unique maximising value $\hat{x} \in \mathbb{R}$ at its sole critical point, and the sign of its first derivative is:

$$\text{sgn} f'(x) = \text{sgn} (\hat{x} - x) \quad \quad \quad \text{for all } x \in \mathbb{R}.$$

In other words, for a continuously differentiable function, strict quasi-concavity means that the derivative is positive for values up to the maximising value (i.e., the slope is increasing up to this point) and then negative after this point (i.e., the slope is decreasing after this point). This is one useful characterisation of quasi-concavity in the case of smooth functions.

Ben
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