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Show that the least natural number a for which x + ax -2 > 2 for all x belonging to (0,infinity) is 2.

Show that the least natural number a for which x + ax -2 > 2  for all x belonging to (0,infinity) is 2.

Grade:12

1 Answers

Ramesh V
70 Points
12 years ago

This can be simply solved using A.M > G.M

as a,x are positive(given conditions0

let the elements be : x/2 , x/2 , a/x2

so A.M = 1/3*[ x/2 + x/2 + a/x2 ]

   G.M = [ x/2 * x/2 * a/x2 ]1/3

so its, x + a/x2 > 3.(a/4)1/3

     as x + a/x2 > 2, it is clear that

        3.(a/4)1/3   >= 2

          a >= 32/27

since a is natural number, min. possible is 2

--

regards

Ramesh

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