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Question

Without using the derivative, show that the function f (x) = | x | is
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0).

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Solution

Here,fx=x(a) Let x1, x2 0, such that x1<x2. Then, x1<x2x1<x2fx1<fx2x1<x2fx1<fx2, x1, x2 0, So, fx is increasing on 0, .(b) Let x1, x2 (-, 0]. such that x1<x2. Then, x1<x2x1>x2fx1>fx2x1<x2fx1>fx2, x1, x2 (-, 0].So, fx is decreasing on (-, 0].

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