Let f ( x ) be a non-negative and upper convex (concave) function defined on the interval [ a , b ]. Suppose f ( a + b 2 ) ≤ 2. Show that f ( x ) ≤ 4 for all x ∈ [ a , b ].

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Let   f  (  x  ) be a non-negative and upper convex (concave) function defined on the interval   [  a  ,  b  ]. Suppose   f      (                  a        +        b            2        )    ≤  2. Show that   f  (  x  )  ≤  4 for all   x  ∈  [  a  ,  b  ].

mangoslush27fig · Accepted Answer

In the case             a      +      b        2    ≤  x  ≤  b we can use the concavity condition for   a  &amp;lt;            a      +      b        2    &amp;lt;  x, which givesNow use that   f  (  a  )  ≥  0 and   x  −  a  ≤  b  −  a.Graphically: Let   l be the line joining   (  a  ,  f  (  a  )  ) and   (            a      +      b        2    ,  f  (            a      +      b        2    )  ). Then   f  (  x  )  ≤  l  (  x  )  ≤  4 for             a      +      b        2    ≤  x  ≤  b.The case   a  ≤  x  ≤            a      +      b        2   works similarly, or can be reduced to the first case because of the symmetry of the problem.

Let f(x) be a non-negative and upper convex (concave) function defined on the interval [a,b]. Suppose f(a+b/2)<=2. Show that f(x)<=4 for all x in [a,b].

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