Question

Let $f:R\to R$ be a function defined by f(x) = Min {x + 1, |x| + 1}. Then which of the following is true

• A. f(x) = 1 for all x R
• B. f(x) is not differentiable at x = 1
• C. f(x) is differentiable everywhere
• D. f(x) is not differentiable at x = 0

Answer 1

The correct option is C f(x) is differentiable everywhere

$f\left(x\right)=Min\{x+1,|x|+1\}$

$x=0\Rightarrow |x|=x$

$f\left(x\right)=min\{x+1,x+1\}$

$=x+1$

$x<0,|x|=-x$

$f\left(x\right)=min\{x+1,-x+1\}$

$-x+1>x+1$

$x<0$

$\Rightarrow x<0,f\left(x\right)=x+1$

$x>0,f\left(x\right)=x+1$

$\Rightarrow f\left(x\right)=x+1$ for all $x$

$\therefore f\left(x\right)$ is differential everywhere (continuous and constant slope)