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  • If x + | y | = 2y , then y as a function of x is.Option: 1 defined for x.Option: 2</stro

If x + | y | = 2y , then y as a function of x is.

Option: 1

defined for x.


Option: 2

Discontinuous at x=0.


Option: 3

differentiable for all x.


Option: 4

such that \mathrm{\frac{d y}{d x}=\frac{1}{3}} for x < 0.


Answers (1)

best_answer

We have, \mathrm{x+\mid y \mid=2 y}
                  \mathrm{ \begin{aligned} \Rightarrow\left\{\begin{array}{cc} x-y=2 y, \text { for } y<0 \\ x+y=2 y, \text { for } y \geq 0 \end{array}\right. \\ \Rightarrow y=\left\{\begin{array}{cc} (1 / 3) x, \text { for } x<0 \\ x, \text { for } x \geq 0 \end{array}\right. \end{aligned} }

Clearly, y = f(x) is continuous for all x but it is not differentiable at x = 0. Because \mathrm{ L f^{\prime}(0)=\frac{1}{3}~ and ~R f^{\prime}(0)=1. ~Also ~\frac{d y}{d x}=f^{\prime}(x)=\frac{1}{3} ~for ~x<0 }.

Posted by

Sanket Gandhi

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