If `f(x)=(x-alpha)^ng(x)` and `f(alpha)=f'(alpha)=f''(alpha)=f^(n-1)(alpha)=0` where f(x) and g(x) are polynomials. For polynomial `f(x)` and `g(x)` with rational cofficients , then answer the following questions (1)If `y = f(x)` touches the x-axis at only one point, then the point of contact

Options

A. sum of the roots of f(x) is 0
B. product of the roots of f(x) is 9
C. sum of the product of the roots taken three at time is `12sqrt(3)`
D. none of these

Answer

To solve the problem step by step, we need to analyze the given function and the conditions provided.

### Step 1: Understand the function
The function is given as:
\[ f(x) = (x - \alpha)^n g(x) \]
where \( g(x) \) is a polynomial. The conditions state that:
\[ f(\alpha) = f'(\alpha) = f''(\alpha) = f^{(n-1)}(\alpha) = 0 \]

### Step 2: Analyze the conditions
The condition \( f(\alpha) = 0 \) indicates that \( \alpha \) is a root of \( f(x) \). The conditions \( f'(\alpha) = 0 \), \( f''(\alpha) = 0 \), and \( f^{(n-1)}(\alpha) = 0 \) imply that \( \alpha \) is a root of multiplicity \( n \).

### Step 3: Touching the x-axis
When a function touches the x-axis at a point, it means that at that point, the function has a root of even multiplicity. Since \( f(x) \) is of the form \( (x - \alpha)^n g(x) \) and \( n \) is the multiplicity, for the function to touch the x-axis at only one point, \( n \) must be even.

### Step 4: Determine the point of contact
Since the function touches the x-axis at \( x = \alpha \), the point of contact is:
\[ \text{Point of contact} = (\alpha, 0) \]

### Step 5: Conclusion
Thus, the point at which the curve \( y = f(x) \) touches the x-axis is:
\[ \boxed{(\alpha, 0)} \]

---