The minimum value of the function $f (x) = ∣ ∣ 2 - | 1 - x | ∣ ∣ - 1,$ where $| x |$ denotes the absolute value of $x,$ is

- 1

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Solution

The correct option is A $- 1$
Clearly, $f (x)$ is defined for all real $x .$
We know that the minimum value of modulus function is zero.
So, $0 \leq ∣ ∣ 2 - | 1 - x | ∣ ∣ < \infty, \forall x \in R$
$\Rightarrow - 1 \leq ∣ ∣ 2 - | 1 - x | ∣ ∣ - 1 < \infty$
$∴$ Minimum value of $f (x) = | 2 - | 1 - x | | - 1$ is $- 1.$

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