Concave function: Difference between revisions

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Revision as of 07:24, 17 June 2007

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A function is called '''quasiconcave''' if and only if there is an <math>x_0</math> such that for all <math>x<x_0</math>, <math>f(x)</math> is non-decreasing while for all <math>x>x_0</math> it is non-increasing. <math>x_0</math> can also be <math>\pm \infty</math>, making the function non-decreasing (non-increasing) for all <math>x</math>. The opposite of quasiconcave is '''quasiconvex'''.

Here's the proof for the fact that the derivative of a concave function is decreasing:

Let x < y < z. Then

u(y) = u�z − y

z − x

x +

y − x

z − x

z��

z − y

z − x

u(x) +

y − x

z − x

u(z)

or equivalently

(z − x)u(y) � (z − y)u(x) + (y − x)u(z) .

This implies immediately

u(y) − u(x)

y − x �

u(z) − u(x)

z − x �

u(z) − u(y)

z − y

.

which is what we wanted to prove

==See also==