Let f(x)`=x^(n+1)+ax^n, "where " a gt 0`. Then, x=0 is point of

Options

A. local minimum for any integer n
B. local minimum if n is an even integer
C. local maximum if n is an even integer
D. local minimum if n is am odd interger

Explanation

To determine whether \( x = 0 \) is a point of local maxima or minima for the function \( f(x) = x^{n+1} + ax^n \) where \( a > 0 \) and \( n > 0 \), we will follow these steps:

### Step 1: Differentiate the function
We start by finding the first derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^{n+1}) + \frac{d}{dx}(ax^n) \]
Using the power rule, we get:
\[ f'(x) = (n+1)x^n + a n x^{n-1} \]

### Step 2: Factor the first derivative
Next, we can factor out \( x^{n-1} \) from the derivative:
\[ f'(x) = x^{n-1} \left((n+1)x + an\right) \]

### Step 3: Evaluate the first derivative at \( x = 0 \)
Now, we evaluate \( f'(x) \) at \( x = 0 \):
\[ f'(0) = 0^{n-1} \left((n+1) \cdot 0 + an\right) = 0 \]
Since \( f'(0) = 0 \), we need to check the behavior of the derivative around \( x = 0 \).

### Step 4: Analyze the sign of the first derivative
To analyze the sign of \( f'(x) \), we consider two cases based on whether \( n \) is even or odd.

#### Case 1: \( n \) is even
If \( n \) is even, then \( n-1 \) is odd. Therefore:
- For \( x < 0 \), \( f'(x) \) will be negative (since \( (n+1)x \) is negative and \( an \) is positive).
- For \( x > 0 \), \( f'(x) \) will be positive (since both terms are positive).

Thus, \( f'(x) \) changes from negative to positive as \( x \) passes through 0, indicating that \( x = 0 \) is a point of local minima.

#### Case 2: \( n \) is odd
If \( n \) is odd, then \( n-1 \) is even. Therefore:
- For \( x < 0 \), \( f'(x) \) will be positive (since \( (n+1)x \) is negative but \( an \) is positive).
- For \( x > 0 \), \( f'(x) \) will still be positive.

In this case, \( f'(x) \) does not change sign as \( x \) passes through 0, indicating that \( x = 0 \) is neither a local maxima nor minima.

### Conclusion
- If \( n \) is even, \( x = 0 \) is a point of local minima.
- If \( n \) is odd, \( x = 0 \) is neither a local maxima nor minima.