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Without using the derivative, show that ...

Without using the derivative, show that the function `f(x)=|x|` is strictly increasing in `(0,oo)` strictly decreasing in `(-oo,0)dot`

Text Solution

Verified by Experts

Here ,
f(x)=|x|
Let `x_{1}, x_{2} in(0, infty)` such that `x_{1} < x_{2}`. Then,
`x_{1} < x_{2}`
`Rightarrow|x_{1}| < |x_{2}|`
`Rightarrow f(x_{1}) < f(x_{2}) `
`therefore x_{1} < x_{2} Rightarrow f(x_{1}) < f(x_{2}), forall x_{1}, x_{2} in(0, infty)`
So, `f(x)` is increasing on `(0, infty)`.
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