If `f(x)` is a polynomial satisfying `f(x)f(1/x)=f(x)+f(1/x)a n df(3)=28 ,t h e nf(4)` is equal to 63 (b) 65 (c) 17 (d) none of these

Options

A. 63
B. 65
C. 17
D. none of these

Explanation

To solve the problem step by step, we need to analyze the given polynomial function \( f(x) \) and use the properties provided in the question.

### Step 1: Understand the given condition
We are given that:
\[ f(x) f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \]
This condition suggests a specific form for the polynomial \( f(x) \).

### Step 2: Identify the form of the polynomial
From the properties of polynomials, we know that a polynomial satisfying the condition \( f(x) f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \) can be expressed as:
\[ f(x) = \pm x^{n+1} \]
where \( n \) is a non-negative integer.

### Step 3: Use the given value \( f(3) = 28 \)
We know:
\[ f(3) = \pm 3^{n+1} = 28 \]
Assuming the positive case:
\[ 3^{n+1} = 28 \]

### Step 4: Solve for \( n \)
To find \( n \), we can express 28 in terms of powers of 3:
\[ 3^{n+1} = 27 \quad \text{(since } 3^3 = 27\text{)} \]
This implies:
\[ n + 1 = 3 \implies n = 2 \]

### Step 5: Write the polynomial
Substituting \( n \) back into the polynomial form:
\[ f(x) = x^{2+1} + 1 = x^3 + 1 \]

### Step 6: Calculate \( f(4) \)
Now we need to find \( f(4) \):
\[ f(4) = 4^3 + 1 \]
Calculating \( 4^3 \):
\[ 4^3 = 64 \]
Thus:
\[ f(4) = 64 + 1 = 65 \]

### Final Answer
The value of \( f(4) \) is:
\[ \boxed{65} \]