1hey i dont understand how have you taken these elements, can you plz explain...
the least natural number 'a' for which x + \frac{a}{x^2} > 2 for all x\epsilon(0,infinity) is
(a) 1
(b) 2
(c) 5
(d) none
ans= (b)
plz explain how?
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3 Answers
11Using A.M > G.M ;
as a,x are positive(given conditions)
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 s
ince a is natural number, min. possible is 2
111It's given x + a/x2 for all x belonging to (0 , ∞)
Divide x into (x/2 + x/2), so we get
(x/2 + x/2 + a/x2) for all x belonging to (0 , ∞)
And then use A/M >G.M