Find maximum number of real solutions for the equation `ax^(n)-bx^(2)+cx-d=0`, when n is a positive even number `(n gt 2)`.

Answer

To find the maximum number of real solutions for the equation \( ax^n - bx^2 + cx - d = 0 \), where \( n \) is a positive even number greater than 2, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Function**:
Let \( f(x) = ax^n - bx^2 + cx - d \).

2. **Differentiate the Function**:
We need to find the first derivative of \( f(x) \):
\[ f'(x) = n \cdot ax^{n-1} - 2bx + c \]

3. **Analyze the Behavior of the Derivative**:
- As \( x \to -\infty \), since \( n \) is even, \( ax^n \) will dominate and \( f'(x) \to +\infty \) (assuming \( a > 0 \)).
- As \( x \to +\infty \), \( f'(x) \to +\infty \) as well (again assuming \( a > 0 \)).

4. **Find Critical Points**:
To find the critical points, set \( f'(x) = 0 \):
\[ n \cdot ax^{n-1} - 2bx + c = 0 \]
This is a polynomial of degree \( n-1 \).

5. **Determine the Number of Critical Points**:
Since \( n \) is even and greater than 2, the polynomial \( f'(x) \) can have up to \( n-1 \) real roots. This means there can be up to \( n-1 \) critical points.

6. **Analyze the Second Derivative**:
The second derivative is:
\[ f''(x) = n(n-1)ax^{n-2} - 2b \]
This will help us determine the nature of the critical points.

7. **Behavior of the Function**:
- Since \( f'(x) \) can have up to \( n-1 \) roots, it can change sign up to \( n-1 \) times. This means \( f(x) \) can have up to \( n-2 \) local extrema (maxima and minima).
- Each local extremum can contribute to the number of real solutions of \( f(x) = 0 \).

8. **Maximum Number of Real Solutions**:
Since \( f(x) \) can have up to \( n-2 \) local extrema, and each of these can contribute to two crossings of the x-axis (one before and one after the extremum), the maximum number of real solutions is:
\[ \text{Maximum Real Solutions} = n - 1 + 1 = n + 1 \]
However, since we are looking for the maximum number of real roots, we conclude that:
\[ \text{Maximum Real Solutions} = n \]

### Conclusion:
Thus, the maximum number of real solutions for the equation \( ax^n - bx^2 + cx - d = 0 \) when \( n \) is a positive even number greater than 2 is \( n \).