At (0,0) ,the curve ` y = x^(1//5)` has

Options

A. a vertical tangent (parallel to y - axis)
B. a horizontal tangent (parallel to x - axis)
C. an oblique tangent
D. no tangent

Answer

To solve the problem, we need to analyze the curve given by the equation \( y = x^{1/5} \) at the point (0,0) and determine the nature of the tangent line at that point.

### Step-by-Step Solution:

1. **Identify the curve**: The curve is given by the equation \( y = x^{1/5} \).

2. **Differentiate the curve**: To find the slope of the tangent line at any point on the curve, we need to calculate the derivative \( \frac{dy}{dx} \).
\[ \frac{dy}{dx} = \frac{d}{dx}(x^{1/5}) = \frac{1}{5}x^{-4/5} \]

3. **Simplify the derivative**: We can rewrite the derivative as:
\[ \frac{dy}{dx} = \frac{1}{5} \cdot \frac{1}{x^{4/5}} = \frac{1}{5x^{4/5}} \]

4. **Evaluate the derivative at the point (0,0)**: We need to find the slope of the tangent line at the point (0,0). Substituting \( x = 0 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{(0,0)} = \frac{1}{5 \cdot 0^{4/5}} \]
This expression results in division by zero, which means the derivative is undefined at this point.

5. **Interpret the result**: Since the derivative is undefined, it indicates that the slope of the tangent line at (0,0) is not defined. This typically suggests that the tangent line is vertical.

6. **Conclusion**: Therefore, at the point (0,0), the curve \( y = x^{1/5} \) has a vertical tangent line.

### Final Answer:
The curve \( y = x^{1/5} \) has a vertical tangent at the point (0,0).

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