Practicing Success
The values of x for which the function $f(x)= \begin{cases}1-x \quad ~~~~, & x<1 \\ (1-x)(2-x), & 1 \leq x \leq 2 \\ 3-x \quad ~~~~, & x>2\end{cases} $ fails to be continuous or differentiable are |
x = 1 x = 2 x = 1, 2 none of these |
x = 2 |
We have, $\lim\limits_{x \rightarrow 1^{-}} f(x)=\lim\limits_{x \rightarrow 1^{-}}(1-x)=0$ $\lim\limits_{x \rightarrow 1^{+}} f(x)=\lim\limits_{x \rightarrow 1^{+}}(1-x)(2-x)=0$ and, f(1) = 0 ∴ $\lim\limits_{x \rightarrow 1^{-}} f(x)=\lim\limits_{x \rightarrow 1^{+}} f(x)=f(1)$ So, f(x) is continuous at x = 1. Now, (LHD at x = 1) = $\left(\frac{d}{d x}(1-x)\right)_{x=1}=-1$ (RHD at x = 1) = $\left(\frac{d}{d x}(1-x)(2-x)\right)_{x=1}=-1$ ∴ (LHD at x = 1) = (RHD at x = 1) So, f(x) is differentiable at x = 1. Now, $\lim\limits_{x \rightarrow 2^{-}} f(x)=\lim\limits_{x \rightarrow 2^{-}}(1-x)(2-x)=0$ and, $\lim\limits_{x \rightarrow 2^{+}} f(x)=\lim\limits_{x \rightarrow 2^{+}} 3-x=1$ ∴ $\lim\limits_{x \rightarrow 2^{-}} f(x) \neq \lim\limits_{x \rightarrow 2^{+}} f(x)$ Thus, f(x) is discontinuous and non-differentiable at x = 2. Hence, the only point of discontinuity and non-differentiability of f(x) is x = 2. |