The curve y `= x^((1)/(5))` has at (0,0)

Options

A. a vertical tangent
B. a horizontal tangent
C. an oblique tangent
D. no tangent

Answer

To determine the nature of the tangent to the curve \( y = x^{\frac{1}{5}} \) at the point (0,0), we will follow these steps:

### Step 1: Differentiate the function
We start with the function:
\[ y = x^{\frac{1}{5}} \]
To find the derivative \( \frac{dy}{dx} \), we apply the power rule of differentiation:
\[ \frac{dy}{dx} = \frac{1}{5} x^{\frac{1}{5} - 1} = \frac{1}{5} x^{-\frac{4}{5}} \]

### Step 2: Evaluate the derivative at the point (0,0)
Next, we need to evaluate the derivative at \( x = 0 \):
\[ \frac{dy}{dx} \bigg|_{x=0} = \frac{1}{5} (0)^{-\frac{4}{5}} \]
Since \( (0)^{-\frac{4}{5}} \) is undefined (as we cannot divide by zero), we conclude that:
\[ \frac{dy}{dx} \bigg|_{x=0} = \infty \]

### Step 3: Determine the nature of the tangent
Since the derivative is infinite at \( x = 0 \), this indicates that the tangent line is vertical at that point. Therefore, the curve \( y = x^{\frac{1}{5}} \) has a vertical tangent at the point (0,0).

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

---