Consider the curve `f(x)=x^(1/3)`, then

Options

A. the equation of tangent at (0, 0) is x = 0
B. the equation of normal at (0, 0) is y = 0
C. normal to the curve does not exist at (0, 0)
D. f(x) and its inverse meet at exactly 3 points

Answer

To solve the problem, we will analyze the curve given by the function \( f(x) = x^{1/3} \) and find the equations of the tangent and normal at the point \( (0, 0) \). We will also check the validity of the statements provided in the question.

### Step-by-Step Solution:

1. **Find the derivative \( f'(x) \)**:
\[ f(x) = x^{1/3} \]
Using the power rule:
\[ f'(x) = \frac{1}{3} x^{-\frac{2}{3}} = \frac{1}{3 \sqrt[3]{x^2}} \]

2. **Evaluate the derivative at the point \( (0, 0) \)**:
\[ f'(0) = \frac{1}{3 \sqrt[3]{0^2}} \text{ is undefined (infinite)} \]
This indicates that the slope of the tangent line at this point is vertical.

3. **Equation of the tangent line**:
Since the slope is infinite, the equation of the tangent line at \( (0, 0) \) is:
\[ x = 0 \]

4. **Find the slope of the normal line**:
The slope of the normal line is the negative reciprocal of the slope of the tangent. Since the slope of the tangent is infinite, the slope of the normal is:
\[ m_n = 0 \]

5. **Equation of the normal line**:
Using the point-slope form of the line equation:
\[ y - 0 = 0 \cdot (x - 0) \implies y = 0 \]

6. **Check the validity of the options**:
- **Option 1**: The equation of the tangent at \( (0, 0) \) is \( x = 0 \) (Correct).
- **Option 2**: The equation of the normal at \( (0, 0) \) is \( y = 0 \) (Correct).
- **Option 3**: The normal to the curve does not exist at \( (0, 0) \) (Incorrect, as we found \( y = 0 \)).
- **Option 4**: \( f(x) \) and its inverse meet at exactly three points. The inverse function \( f^{-1}(x) = x^3 \). Setting \( f(x) = f^{-1}(x) \):
\[ x^{1/3} = x^3 \implies x^9 = x \implies x(x^8 - 1) = 0 \]
This gives us \( x = 0 \) and \( x = 1, -1 \) (three solutions). Thus, this option is also correct.

### Final Conclusion:
- The equations of the tangent and normal at the point \( (0, 0) \) are \( x = 0 \) and \( y = 0 \) respectively.
- The statements regarding the tangent and normal are correct except for the one stating that the normal does not exist.