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Question
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
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Solution
We have `f (x) = x + 1/x, x in I`
Differentiating w.r.t.x, we get
`f' (x) = 1 - 1/x^2 = (x^2 - 1)/x^2`
`x^2 > 0 (1, 1), x^2 - 1 > 0 = x^2 > 1`
= `x < - 1 or x > 1`
= `x in (-oo, -1) or x in (1, oo)`
= `x in (-oo, -1) cup (1, oo) `
= `x in R - (-1, 1)`
= f (x) is strictly increasing on I
(∵ I is an interval which is a subset of R - (-1, 1))
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