Question

Let I be any interval disjoint from (−1, 1). Prove that the function f given by is strictly increasing on I.

Answer 1

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.

solution :- f(x) = x + 1/x
differentiate with respect to x
f'(x) = 1 - 1/x²
for strictly increasing , f'(x) > 0
so, f'(x) = 1 - 1/x² > 0
=> (x² - 1)/x² > 0
=> (x - 1)(x + 1)/x² > 0
because x² ≠ 0 and x² > 0 for all real numbers
so, (x - 1)(x + 1) > 0
x > 1 or x < -1
hence, f(x) is strictly increasing on x ∈ (1 , ∞) U (-∞, -1)
hence, f(x) is strictly increasing on disjoint on I

[note :- if (a,b) is interval then (-∞,a) U (b,∞) is known as disjoint ]