If the only point of inflection of the function `f(x)-(x-a)^m(x-b)^n` , `m, n e N` and `m!=n` is at `x=a` then

Options

A. m, n are even
B. m is odd n is even
C. m is even, n is odd
D. m, n are odd

Answer

To solve the problem, we need to analyze the function \( f(x) = (x-a)^m (x-b)^n \) where \( m, n \in \mathbb{N} \) and \( m \neq n \). We are given that the only point of inflection is at \( x = a \).

### Step-by-Step Solution:

1. **Understanding Points of Inflection**:
A point of inflection occurs where the second derivative of the function changes sign. This means we need to find the second derivative \( f''(x) \) and determine where it equals zero.

2. **Finding the First Derivative \( f'(x) \)**:
We can use the product rule to find the first derivative:
\[ f'(x) = \frac{d}{dx}[(x-a)^m] \cdot (x-b)^n + (x-a)^m \cdot \frac{d}{dx}[(x-b)^n] \]
Using the chain rule:
\[ f'(x) = m(x-a)^{m-1}(x-b)^n + n(x-a)^m(x-b)^{n-1} \]

3. **Finding the Second Derivative \( f''(x) \)**:
We differentiate \( f'(x) \) again using the product rule:
\[ f''(x) = \frac{d}{dx}[m(x-a)^{m-1}(x-b)^n] + \frac{d}{dx}[n(x-a)^m(x-b)^{n-1}] \]
This results in a more complex expression. We will simplify it later.

4. **Setting Up for Inflection Point**:
For \( x = a \) to be the only point of inflection, we need \( f''(a) = 0 \) and \( f''(x) \) must not change sign for any \( x \neq a \).

5. **Evaluating \( f''(a) \)**:
Substitute \( x = a \) into \( f''(x) \):
\[ f''(a) = m(m-1)(a-b)^n + n(n-1)(a-a)^{m-1}(a-b)^{n-1} + \text{other terms} \]
The term \( (a-a)^{m-1} \) will vanish, leading to:
\[ f''(a) = m(m-1)(a-b)^n \]

6. **Conditions for Inflection**:
For \( f''(a) = 0 \), we need \( m(m-1)(a-b)^n = 0 \). Since \( a \neq b \), we conclude \( m(m-1) = 0 \). Thus, \( m = 1 \).

7. **Analyzing \( n \)**:
Since \( m \neq n \) and \( m = 1 \), \( n \) must be greater than 1. Therefore, \( n \) can be any natural number greater than 1.

8. **Conclusion**:
The only point of inflection at \( x = a \) implies that \( m = 1 \) and \( n \) must be any natural number greater than 1.

### Final Answer:
Thus, the values of \( m \) and \( n \) are:
- \( m = 1 \)
- \( n \in \mathbb{N}, n > 1 \)