Show that g ( x ) = x ln ⁡ x and g ( x ) = e x are bounded below.Show that g ( x ) is bounded below, for 0 ≤ xa) g ( x ) = { 0 if x = 0 x ln ⁡ x if x &gt; 0 b) g ( x ) = e x For (a), g ( x ) ≥ 0 for x ≥ 0For (b), g ( x ) ≥ 1 for x ≥ 0Is that all I have to say? Or is there a more technical definition/proof?

Question

Show that   g  (  x  )  =  x  ln  ⁡      x   and   g  (  x  )  =      e    x   are bounded below.Show that   g  (  x  ) is bounded below, for   0  ≤  xa)   g  (  x  )  =      {                            0                                      if                     x          =          0                                      x          ln          ⁡                      x                                                if                     x          &amp;gt;          0                        b)   g  (  x  )  =      e    x  For (a),   g  (  x  )  ≥  0 for   x  ≥  0For (b),   g  (  x  )  ≥  1 for   x  ≥  0Is that all I have to say? Or is there a more technical definition/proof?

Julius Haley · Accepted Answer

Your answer is correct. The inverse Laplace transform is      L    {  F  (  s  )  }  =      1          2      π      i            ∫          γ      −      i      ∞              γ      +      i      ∞        F  (  s  )      e          s      t        d  s  .In your problem, we have that   F  (  s  )  =      1          s      −      λ       so we have a simple pole in the s plane at   s  =  λ                              1                      2            π            i                                    ∫                      γ            −            i            ∞                                γ            +            i            ∞                          F        (        s        )                  e                      s            t                          d        s                            =                  1                      2            π            i                                    ∫                      γ            −            i            ∞                                γ            +            i            ∞                                                e                          s              t                                            s            −            λ                          d        s                                          =        ∑        Res                                          =                  lim                      s            →            λ                          (        s        −        λ        )                              e                          s              t                                            s            −            λ                                                            =                  e                      t            λ                              as you have found.

Show that g(x)=x ln(x) and g(x)=e^x are bounded below.

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