Limit Continuity And Differentiability Ques 35
- If $x+|y|=2 y$, then $y$ as a function of $x$ is
$(1984,2 M)$
(a) defined for all real $x$
(b) continuous at $x=0$
(c) differentiable for all $x$
(d) such that $\frac{d y}{d x}=\frac{1}{3}$ for $x<0$
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Answer:
Correct Answer: 35.(a, b, d)
Solution: (a, b, d) Since, $x+|y|=2 y \Rightarrow \begin{cases}x+y=2 y, & \text { when } y>0 \\ x-y=2 y, & \text { when } y<0\end{cases}$
$\Rightarrow \quad\left\{\begin{array}{cc}y=x, & \text { when } y>0 \Rightarrow x>0 \\ y=x / 3, & \text { when } y<0 \Rightarrow x<0\end{array}\right.$
which could be plotted as,
Clearly, $y$ is continuous for all $x$ but not differentiable at $x=0$.
Also, $\quad \frac{d y}{d x}=\left\{\begin{array}{cc}1, & x>0 \\ 1 / 3, & x<0\end{array}\right.$
Thus, $f(x)$ is defined for all $x$, continuous at $x=0$, differentiable for all $x \in R-\{0\}, \frac{d y}{d x}=\frac{1}{3}$ for $x<0$.
BETA
