If `x+4|y|=6y`, then y as a function of x is

Options

A. continuous at x=0
B. derivable at x=0
C. `(dy)/(dx)=(1)/(2) ` for all x
D. none of these

Answer

To solve the equation \( x + 4|y| = 6y \) and express \( y \) as a function of \( x \), we need to consider the different cases for \( y \) based on the absolute value.

### Step-by-Step Solution:

1. **Identify Cases for \( |y| \)**:
The absolute value \( |y| \) can be expressed in two cases:
- Case 1: \( y \geq 0 \) (i.e., \( |y| = y \))
- Case 2: \( y < 0 \) (i.e., \( |y| = -y \))

2. **Case 1: \( y \geq 0 \)**:
Substitute \( |y| = y \) into the equation:
\[ x + 4y = 6y \]
Rearranging gives:
\[ x = 6y - 4y \implies x = 2y \]
Therefore, we can express \( y \) in terms of \( x \):
\[ y = \frac{x}{2} \]

3. **Case 2: \( y < 0 \)**:
Substitute \( |y| = -y \) into the equation:
\[ x + 4(-y) = 6y \]
This simplifies to:
\[ x - 4y = 6y \implies x = 6y + 4y \implies x = 10y \]
Therefore, we can express \( y \) in terms of \( x \):
\[ y = \frac{x}{10} \]

4. **Summary of Cases**:
- If \( y \geq 0 \), then \( y = \frac{x}{2} \).
- If \( y < 0 \), then \( y = \frac{x}{10} \).

5. **Determine Continuity at \( x = 0 \)**:
- For \( x = 0 \):
- From \( y = \frac{x}{2} \), we get \( y = 0 \).
- From \( y = \frac{x}{10} \), we also get \( y = 0 \).
- Thus, \( y \) is continuous at \( x = 0 \).

6. **Check Differentiability**:
- The derivative \( \frac{dy}{dx} \) for \( y = \frac{x}{2} \) is \( \frac{1}{2} \).
- The derivative \( \frac{dy}{dx} \) for \( y = \frac{x}{10} \) is \( \frac{1}{10} \).
- Since the derivatives from the left and right do not match at \( x = 0 \), \( y \) is not differentiable at \( x = 0 \).

### Final Answer:
Thus, \( y \) as a function of \( x \) is given by:
\[ y = \begin{cases} \frac{x}{2} & \text{if } x \geq 0 \\ \frac{x}{10} & \text{if } x < 0 \end{cases} \]