How to geometrically prove that x + 1/x >=2, if x>0
Answers
solution
take x = 1 :— 1 + (1/1) = 2 >=2
take x = 2 :— 2 + (1/2) = 2.5 >= 2
and so on.
anything bigger than 1 when added to anything — no matter how small it is, will make equal to or more than 2 for obvious reasons.
But This kind of answer may get you a zero in exams.
let’s solve it in bookish manner —
x + (1/x) >= 2 (assuming this to be true)
=> x^2 + 1 >= 2x
=> x^2 -2x +1 >= 0
=> (x-1)^2 >= 0
=> (x-1)(x-1)>=0
=> x>=1
Hence we can see that the inequality is true only for the positive integers bigger than 1.
Answer:
Step-by-step explanation:
First make the equation into x^2-2x+1 >= 0 by rearranging and multiplying by x. You should find it is (x-1)^2 so for any x (not just x>0) this function is greater than or equal to 0 because of ^2. For your situation it is obvious that x>0 which is certainly true for the factorised function. Therefore you have proved that x+1/x is greater than or equal to 2.
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