Let D be a convex set in R n and f : D → R a concave and C 1 function. How do I show that x ∗ is a global maximum for f if and only if f ( 1 ) ( x ∗ ) y ≤ 0 for all y pointing into D at x ∗ (Here f ( 1 ) denotes the first derivative of f)

Question

Let   D be a convex set in             R        n   and   f  :  D  →      R   a concave and       C    1   function. How do I show that       x    ∗   is a global maximum for   f if and only if       f          (      1      )        (      x    ∗    )  y  ≤  0 for all   y pointing into   D at       x    ∗   (Here       f          (      1      )       denotes the first derivative of   f)

Raven Hawkins · Accepted Answer

If we had       f          (      1      )        (      x    ∗    )  y  &amp;gt;  0 then by the definition of the derivative we could find a point close by in that direction for which the value   f is higher. That shows the only if part.For the other direction, take any other point   y  ∈  D. Consider a straight path between the two points   v  (  t  )  :  [  0  ,  1  ]  →            R        d  ,   v  (  t  )  :  [  0  ,  1  ]  →            R        d  .   f  (  v  (  t  )  ) is concave and continuously differentiable with respect to   t, so must have nonincreasing derivative. Combining this observation with the fact that       f          (      1      )        (      x    ∗    )  y  ≤  0 and the fundamental theorem of calculus, we see that   f  (  y  )  ≤  f  (      x    ∗    ).

Let D be a convex set in R^n and f:D->R a concave and C^1 function. How do we show that x^∗ is a global maximum for f if and only if f^(1)(x∗)y<=0 for all y pointing into D at x^∗ (Here f^(1) denotes the first derivative of f)

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New Questions in Calculus 1