Let I ⊂ R ≥ 0 be an interval and let f : I → R ≥ 0 be concave (and smooth enough). Weather the following inequality holds: f ( a + b ) ≤ f ( a ) + f ( b ) .Is this true?

Question

Let   I  ⊂            R              ≥      0       be an interval and let   f  :  I  →            R              ≥      0       be concave (and smooth enough). Weather the following inequality holds:  f  (  a  +  b  )  ≤  f  (  a  )  +  f  (  b  )  .Is this true?

Kristen Garza · Accepted Answer

It is true with   f  (  x  )  =      x          1              /            p      ,   1  ≤  p  &amp;lt;  ∞. To see that, note that we just have to deal with the case   b  =  1. Then put   g  (  x  )  :=      x          1              /            p        +  1  −  (  1  +  x      )          1              /            p       and show that this function is non-decreasing.In the general case, if   0  ∈  I and   f  (  0  )  =  0 it&#039;s true.

Let I sub R^(>=0) be an interval and let f:I->R^(>=0) be concave (and smooth enough). Weather the following inequality holds: f(a+b)<=f(a)+f(b). Is this true?

Answered question

Answer & Explanation

New Questions in Calculus 1