Without using the Second Derivative Test, can the convexity of the natural exponential function be shown directly from the definition of convexity? The expression e t = ∑ n = 0 ∞ t n n ! can be used.

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Without using the Second Derivative Test, can the convexity of the natural exponential function be shown directly from the definition of convexity? The expression      e          t        =      ∑          n      =      0              ∞                  t              n                    n      !      can be used.

Siffkiffanewsler6n · Accepted Answer

Step 1Demonstration in the case that   x  &amp;gt;  0 ,   y  &amp;gt;  0 , and   s  ,    t  ∈            Q              +      Since s is a positive, proper, rational number, there are positive integers   j  &amp;lt;  k such that   s  =  j      /    k , and since s and t are a pair of conjugate numbers,   t  =  (  k  −  j  )      /    k . For any positive, real numbers       z          1        ,        z          2        ,    …        z          n       , and for any positive integer n,            (                                    z                          1                                +                      z                          2                                +          …          +                      z                          k                                      k            )              n        ≤                                          z                          1                                                n                    +                                    z                          2                                                n                    +      …      +                                    z                          k                                                n                      k    . .If z      z          1        =      z          2        =  …  =      z          j        =  x and       z          j      +      1        =      z          j      +      2        =  …  =      z          k        =  y ,                    (        s        x        +        t        y                  )                      n                                              =                              (                          (                              j                k                            )                        x            +                          (                                                k                  −                  j                                k                            )                        y            )                                n                                                            =                              (                                          j                x                +                (                k                −                j                )                y                            k                        )                                n                                                            ≤                              j                          x                              n                                      +            (            k            −            j            )                          y                              n                                              k                                                  =                  (                      j            k                    )                          x                      n                          +                  (                                    k              −              j                        k                    )                          y                      n                                                            =        s                  x                      n                          +        t                  y                      n                          .             .Consequently,            (      s      x      +      t      y              )                  n                            n      !        ≤            s              x                  n                    +      t              y                  n                            n      !        . .

Without using the Second Derivative Test, can the convexity of the natural exponential function be s

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