Let f be a concave function (and differentiable). Show that f ( y ) − f ( x ) y − x ≥ f ′ ( y )where y &gt; x.

Question

Let   f be a concave function (and differentiable). Show that            f      (      y      )      −      f      (      x      )              y      −      x        ≥      f    ′    (  y  )where   y  &amp;gt;  x.

Jovanni Salinas · Accepted Answer

You case use Lagrange&#039;s theorem.            f      (      y      )      −      f      (      x      )              y      −      x        =      f    ′    (  ξ  )  ,    ξ  ∈  [  x  ,  y  ]Since   f is concave,       f    ′   is non-increasing and so   ξ  ≤  y  ⇒      f    ′    (  ξ  )  ≥      f    ′    (  y  ). As it was pointed out in another answer, the previous conclusion depends on the fact that   x  &amp;lt;  y.This gives you the desired inequality.

Let f be a concave function (and differentiable). Show that (f(y)−f(x))/(y−x)>=f′(y)

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