Let `f(x)=x + 2|x+1|+2| x-1|`. Find the values of `k` if `f(x)=k`
(i) has exactly one real solution,
(ii) has two negative solutions,
(iii) has two solutions of opposite sign.

Explanation

To solve the problem, we need to analyze the function \( f(x) = x + 2|x + 1| + 2|x - 1| \) and determine the values of \( k \) for different conditions regarding the number of solutions to the equation \( f(x) = k \).

### Step 1: Analyze the function \( f(x) \)

The function \( f(x) \) involves absolute values, which means it will behave differently in different intervals based on the critical points where the expressions inside the absolute values change sign. The critical points are \( x = -1 \) and \( x = 1 \). We will evaluate \( f(x) \) in the following intervals:

1. \( x < -1 \)
2. \( -1 \leq x < 1 \)
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