Question

If x + | y | = 2y, th e n y as a function of x is

• A. defined for all real x
• B. continuous at x = 0
• C. differentiable for all x
• D. such that $\frac{dy}{dx} = \frac{1}{3}$ for x < 0

Answer 1

Answer: (D)

such that $\frac{dy}{dx} = \frac{1}{3}$ for x < 0

Answer 2

Since, x + | y |= 2 y \Rightarrow \bigg \\{ \begin{array} \ x + y = 2y, \\\ x - y = 2y, \\\ \end{array} \begin{array} when y > 0 \\\ when y < 0 \\\ \end{array}
\Rightarrow \bigg \\{ \begin{array} \ y = x, \ when \ y > 0 \Rightarrow x > 0 \\\ y = x/3, \ when \ y < 0 \Rightarrow x < 0 \\\ \end{array}
which could be plotted as,
Clearly, y is continuous for all x but not differentiable at x = 0,
Also, \frac{dy}{dx} = \bigg \\{ \begin{array} \ 1, \ x > 0 \\\ 1/3, \ x < 0 \\\ \end{array}
Thus, f (x) is defined for all x, continuous at x = 0,
differentiable for all x $\in \ R - \\{ 0 \\}, \frac{dy}{dx}, = \frac{1}{3}$ for x < 0.