Maximum number of real solution for the equation
`ax^(n)+x^(2)+bx+c=0,"where "a,b,c in R ` and n is an even positive number, is

Options

A. 2
B. 3
C. 4
D. infinite

Answer

To determine the maximum number of real solutions for the equation \( ax^n + x^2 + bx + c = 0 \), where \( a, b, c \in \mathbb{R} \) and \( n \) is an even positive integer, we can follow these steps:

### Step 1: Identify the Degree of the Polynomial
The given equation is a polynomial of degree \( n \) (the term \( ax^n \) dominates), where \( n \) is an even positive integer.

**Hint:** The degree of the polynomial determines the maximum number of real solutions.

### Step 2: Understand the Properties of Even Degree Polynomials
For any polynomial of even degree, the graph of the polynomial will approach positive or negative infinity as \( x \) approaches positive or negative infinity. This means that the polynomial can cross the x-axis multiple times.

**Hint:** Even degree polynomials can have a maximum of \( n \) real roots.

### Step 3: Analyze the Coefficients
The coefficients \( a, b, c \) can be any real numbers. The leading coefficient \( a \) (associated with \( x^n \)) will determine the end behavior of the polynomial. If \( a > 0 \), the polynomial will rise to infinity on both ends; if \( a < 0 \), it will fall to negative infinity on both ends.

**Hint:** The sign of the leading coefficient affects the number of real solutions.

### Step 4: Determine the Maximum Number of Real Solutions
Since \( n \) is an even positive integer, the maximum number of real solutions (roots) that the polynomial can have is equal to \( n \). However, since \( n \) can be any even positive integer (2, 4, 6, ...), theoretically, there is no upper limit to the number of solutions.

**Hint:** The maximum number of real solutions is determined by the value of \( n \), which can be infinitely large.

### Conclusion
Thus, the maximum number of real solutions for the equation \( ax^n + x^2 + bx + c = 0 \) is infinite, as \( n \) can take any even positive integer value.

**Final Answer:** The maximum number of real solutions is infinite.