If `f(x)=x+(1)/(x)`, such that `f^3 (x)=f(x^(3))+lambdaf((1)/(x))`, then `lambda=`

Options

A. 1
B. 3
C. -3
D. -1

Explanation

To solve the problem, we need to find the value of \(\lambda\) in the equation given by:

\[ f^3(x) = f(x^3) + \lambda f\left(\frac{1}{x}\right) \]

where \(f(x) = x + \frac{1}{x}\).

### Step 1: Calculate \(f^3(x)\)

First, we need to compute \(f^3(x)\):

\[ f^3(x) = \left(f(x)\right)^3 = \left(x + \frac{1}{x}\right)^3 \]

Using the binomial expansion formula \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\), we can expand this:

\[ f^3(x) = x^3 + \left(\frac{1}{x}\right)^3 + 3\left(x\right)\left(\frac{1}{x}\right)\left(x + \frac{1}{x}\right) \]

Calculating the individual components:

- \(x^3\)
- \(\left(\frac{1}{x}\right)^3 = \frac{1}{x^3}\)
- \(3\left(x\right)\left(\frac{1}{x}\right) = 3\)

Thus, we have:

\[ f^3(x) = x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) \]

This can be rewritten as:

\[ f^3(x) = x^3 + \frac{1}{x^3} + 3f(x) \]

### Step 2: Calculate \(f(x^3)\) and \(f\left(\frac{1}{x}\right)\)

Next, we calculate \(f(x^3)\):

\[ f(x^3) = x^3 + \frac{1}{x^3} \]

Now, we calculate \(f\left(\frac{1}{x}\right)\):

\[ f\left(\frac{1}{x}\right) = \frac{1}{x} + x = x + \frac{1}{x} = f(x) \]

### Step 3: Substitute into the equation

Now we substitute \(f(x^3)\) and \(f\left(\frac{1}{x}\right)\) into the right-hand side of the original equation:

\[ f^3(x) = f(x^3) + \lambda f\left(\frac{1}{x}\right) \]

Substituting the values we found:

\[ x^3 + \frac{1}{x^3} + 3f(x) = x^3 + \frac{1}{x^3} + \lambda f(x) \]

### Step 4: Compare both sides

Now, we can compare both sides of the equation:

The terms \(x^3 + \frac{1}{x^3}\) cancel out from both sides:

\[ 3f(x) = \lambda f(x) \]

### Step 5: Solve for \(\lambda\)

Assuming \(f(x) \neq 0\), we can divide both sides by \(f(x)\):

\[ 3 = \lambda \]

Thus, the value of \(\lambda\) is:

\[ \lambda = 3 \]

### Final Answer

\(\lambda = 3\)