Show that the function f (x) = |x| is strictly increasing on ] 0, infinity[ - Maths - Application of Derivatives

Question

Show that the function f (x) = |x| is strictly increasing on ]
0, infinity[

Priyanka Kedia · Accepted Answer

$A f u n c t i o n i s s t r i c t l y i n c r e a \sin g i f x_{1} < x_{2} \Rightarrow f (x_{1}) < f (x_{2}) L e t x_{1}, x_{2} b e t w o p o s i t i v e r e a l n u m b e r s u c h t h a t, x_{1} < x_{2} x_{1} > 0 \Rightarrow {x_{1}}^{2} < x_{1} x_{2} ...... (1) x_{2} > 0 \Rightarrow x_{1} x_{2} {< x_{2}}^{2} ...... (2) F r o m (1) a n d (2), {x_{1}}^{2} < {x_{2}}^{2} x_{1} < x_{2} (\sin c e, x_{1}, x_{2} b e t w o p o s i t i v e r e a l n u m b e r) T h u s, x_{1} < x_{2} \Rightarrow f (x_{1}) < f (x_{2}) (f r o m f i g u r e) H e n c e, f (x) = | x | i s s t r i c t l y i n c r e a \sin g f u n c t i o n o n (0, \infty)$