Q168 / 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 continuous but not differentiable at $\mathrm{x}=0$
- Correct answerB$f(x)$ is continuous but not differentiable at $x=1$
- C$\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=1$
- D$\mathrm{f}(\mathrm{x})$ is not differentiable at $\mathrm{x}=0$ and $\mathrm{x}=1$
Solution
$\begin{aligned}
& f(x)=|x-1|+x^2 \quad \forall \quad x \in R \\
& f_1(x)=|x-1|, f_2(x)=x^2
\end{aligned}$
$f_1(x)$ and $f_2(x)$ both are continuous. Hence $f(x)$ is continuous.
$f(x)$ in differentiable at $x=0$
$f_1(x)$ is not differentiable at $x=1$.
Hence $f(x)$ is continuous but not differentiable at $x=1$.
Hence, the correct answer is option: 'B'.