Given a continuous function f : [ 0 , 1 ] → R , prove that ∀ t &gt; 0 , 1 t ⋅ ln ⁡ ( ∫ 0 1 e − t f ( x ) d x ) ≤ − min f ( x ) .

Question

Given a continuous function   f  :  [  0  ,  1  ]  →      R  , prove that  ∀  t  &amp;gt;  0  ,      1    t    ⋅  ln  ⁡      (          ∫              0                    1                    e              −        t        f        (        x        )              d    x    )    ≤  −  min  f  (  x  )  .

lofoptiformfp · Accepted Answer

The inequality says      ∫    0    1        e          −      t      f      (      x      )          d  x  ≤      e          −      t              min        x            f      (      x      )        .Since   t  &amp;gt;  0  ,, the quantity       e          −      t              min        x            f      (      x      )       is the largest value of the function       e          −      t      f      (      x      )       within   0  ≤  x  ≤  1.. So this just says that      ∫    0    1    g  (  x  )    d  x  ≤  (  1  −  0  )  ×  max  {  g  (  x  )  :  0  ≤  x  ≤  1  }  .

Given a continuous function f : [ 0 , 1 ] → <mrow class

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