The set of all a for which the equation x | x - 1 | + | x + 2 | + a = 0 has exactly one real root, is [2023]

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Solution

Q.
The set of all $a \in ℝ$ for which the equation $x$ $| x - 1 |$ $+$ $| x + 2 |$ $+ a = 0$ has exactly one real root, is [2023]

1 $(- \infty, \infty)$

2 $(- 6, \infty)$

3 $(- \infty, - 3)$

4 $(- 6, - 3)$

Ans.
(1)

Let $f (x) = x$ $| x - 1 |$ $+$ $| x + 2 |$

Given, $x$ $| x - 1 |$ $+$ $| x + 2 |$ $+ a = 0$

$\Rightarrow f (x) + a = 0 \Rightarrow - a = f (x)$

$f (x) =$ ${\begin{matrix} - x^{2} - 2, & - \infty < x < - 2 \\ - x^{2} + 2 x + 2, & - 2 \leq x < 1 \\ x^{2} + 2, & 1 \leq x < \infty \end{matrix}$

All values are increasing.

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