# Find the intervals in which f(x)=|x-2|÷x² is strictly increasing and strictly decreasing

8 years ago
$f(x)=\frac{|x-2|}{x^{2}}$
$f(x)=\frac{x-2}{x^{2}} for x>2$

$f(x)=\frac{2-x}{x^{2}} for x<2$

$f'(x)=\frac{4}{x^{3}}-\frac{1}{x^{2}} {\color{Blue} for x>2 }$
$f'(x)=-\frac{4}{x^{3}}+\frac{1}{x^{2}} {\color{Blue} for x<2 }$

for strictly increasing f'(x)>0
$f'(x)=\frac{4}{x^{3}}-\frac{1}{x^{2}}>0 {\color{Blue} for x>2 }$
$f'(x)=-\frac{4}{x^{3}}+\frac{1}{x^{2}}>0 {\color{Blue} for x<2 }$
$x\varepsilon (0,2)\bigcup (4,\varpi )$

for strictly increasing f'(x)<0
$f'(x)=-\frac{4}{x^{3}}+\frac{1}{x^{2}}<0 {\color{Blue} for x<2 }$
$f'(x)=\frac{4}{x^{3}}-\frac{1}{x^{2}}<0 {\color{Blue} for x>2 }$
$x\varepsilon (-\varpi,0)\bigcup (2,4 )$

B.Tech IIT Delhi.