Q169 / 270 · NDA 2016 (Phase 1) · Mathematics
Consider the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}-1|+\mathrm{x}^2$, where $\mathrm{x} \in \mathrm{R}$.
Which one of the following statements is correct?
- A$\mathrm{f}(\mathrm{x})$ is increasing in $\left(-\infty, \frac{1}{2}\right)$ and decreasing in $\left(\frac{1}{2}, \infty\right)$
- Correct answerB$f(x)$ is decreasing in $\left(-\infty, \frac{1}{2}\right)$ and increasing in $\left(\frac{1}{2}, \infty\right)$
- C$f(x)$ is increasing in $(-\infty, 1)$ and decreasing in $(1, \infty)$
- D$f(x)$ is decreasing in $(-\infty, 1)$ and increasing in $(1, \infty)$
Solution
As we know,
$\begin{aligned}
& f(x)=|x-1|+x^2 \quad \forall x \in R \\
& f(x)= \begin{cases}x-1+x^2 & x \geq 1 \\
1-x+x^2 & x \lt 1\end{cases} \\
& \Rightarrow f^{\prime}(x)= \begin{cases}2 x+1 ; & x \geq 1 \\
2 x-1 ; & x \lt 1\end{cases}
\end{aligned}$
$f(x)$ is in quadratic form (parabola). Hence $f(x)$ is decreasing in $\left(-\infty, \frac{1}{2}\right)$ and increasing $\left(\frac{1}{2}, \infty\right)$.
Hence, the correct answer is option: 'B'.