Question

The maximum value of $f\left(x\right)=100-|45-x|$ for any value of $x$ is $k-x$ then $k$ is ____.

Answer 1

Given $f\left(x\right)=100-|45-x|$

Case 1- $45-x>0$

$\therefore |45-x|=45-x$

$\therefore f\left(x\right)=100-(45-x)$

$\therefore f\left(x\right)=100-45+x$

$\therefore y=f\left(x\right)=55+x$ (1)

Case 2- $45-x<0$

$\therefore |45-x|=-(45-x)$

$\therefore |45-x|=x-45$

$\therefore f\left(x\right)=100-(x-45)$

$\therefore f\left(x\right)=100-x+45$

$\therefore y=f\left(x\right)=145-x$ (2)

Equate equations (1) and (2),

$55+x=145-x$

$\therefore 2x=145-55$

$\therefore 2x=90$

$\therefore x=45$

Thus, maximum function is given by,

$f\left(x\right)=55+x$ for $x>45$

$f\left(x\right)=145-x$ for $x<45$