Discussion Forum

 The function f(x) = 2|x|+|x+2|-||x+2|-2|x||  has a local minimum or a local maximum at x is equal to 

Answer:

Explanation:

Concept involved

 We know, 

$|x|=\begin{cases}x & x \geq 0\\-x, & x < 0\end{cases}$
$\Rightarrow$      $|x-a|=\begin{cases}x-a & x \geq0\\-(x-a) & x < a\end{cases}$

 and for non-differentiable continuous function the maximum or minimum Can be checked with graph as

Here  f(x) =2|x|+|x+2| -||x+2|-2|x||

$=\begin{cases}-2x-(x+2)+(x+2), & when x\leq-2\\-2x+x+2+3x+2, & when -2<x\leq-2/3\\ -4x&when- \frac{2}{3}<x\leq0\\ 4x,&when 0<x\leq2\\ 2x+4,&when x>2\end{cases}$

 $=\begin{cases}-2x-4, &  x\leq-2\\2x+4, &  -2<x\leq-2/3\\ -4x&- \frac{2}{3}<x\leq0\\ 4x,& 0<x\leq2\\ 2x+4,& x>2\end{cases}$

 Graph for y=f(x) is shown as