Let f : Ω ⊆ R n → R ≥ 0 be a continuous differentiable function over Ω. Suppose that the function f is concave, and fix two points x = ( x 1 , … , x n ) , y = ( y 1 , … , y n ) ∈ Ω, x = ( x 1 , … , x n ) , y = ( y 1 , … , y n ) ∈ Ω.If x i ≤ y i for all i = 1 , … , n and Ω = R n , does it hold ∥ ∇ x f ∥≥∥ ∇ y f ∥?

Question

Let   f  :  Ω  ⊆            R        n    →            R              ≥      0       be a continuous differentiable function over   Ω. Suppose that the function   f is concave, and fix two points       x    =  (      x    1    ,  …  ,      x    n    )  ,      y    =  (      y    1    ,  …  ,      y    n    )  ∈  Ω,      x    =  (      x    1    ,  …  ,      x    n    )  ,      y    =  (      y    1    ,  …  ,      y    n    )  ∈  Ω.If       x    i    ≤      y    i   for all   i  =  1  ,  …  ,  n and   Ω  =            R        n  , does it hold   ∥      ∇                  x              f  ∥≥∥      ∇                  y              f  ∥?

sliceu4i · Accepted Answer

Is false. Take a concave function which is symmetric about the origin (e.g.   f  =  −  ‖  x      ‖    2  ). if   0  ≤      x    i   and   ‖  ∇  f  (      x    )  ‖  &amp;gt;  ‖  ∇  f  (  0  )  ‖, then due to symmetry we&#039;d get   −      x    i    ≤  0 but   ‖  ∇  f  (  −      x    )  ‖  &amp;lt;  ‖  ∇  f  (  0  )  ‖.

Let f:Omega sube R^n->R_(>=0) be a continuous differentiable function over Omega. Suppose that the function f is concave, and fix two points x=(x_1,…,x_n),y=(y_1,…,y_n) in Omega. If x_i≤y_i for all i=1,…,n and Omega=R^n, does it hold ∥grad_xf∥>=∥grad_yf∥?

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New Questions in Calculus 1