If ` f(x) =( lamda^2-1)/( lamda^2 +1) x^3 - 3x +5 ` is a decreasing function of `lamda`(independent of x) is

Options

A. `[-1,1]`
B. ` (-oo,-1)`
C. `(1,oo)`
D. None

Answer

To determine the values of \( \lambda \) for which the function

\[ f(x) = \frac{\lambda^2 - 1}{\lambda^2 + 1} x^3 - 3x + 5 \]

is a decreasing function independent of \( x \), we need to analyze the derivative of \( f(x) \) with respect to \( x \).

### Step 1: Differentiate \( f(x) \)

The derivative \( f'(x) \) is given by:

\[ f'(x) = \frac{d}{dx} \left( \frac{\lambda^2 - 1}{\lambda^2 + 1} x^3 - 3x + 5 \right) \]

Using the power rule, we differentiate each term:

\[ f'(x) = \frac{\lambda^2 - 1}{\lambda^2 + 1} \cdot 3x^2 - 3 \]

Thus, we have:

\[ f'(x) = \frac{3(\lambda^2 - 1)}{\lambda^2 + 1} x^2 - 3 \]

### Step 2: Set \( f'(x) \leq 0 \)

For \( f(x) \) to be a decreasing function, we need:

\[ f'(x) \leq 0 \]

This leads to the inequality:

\[ \frac{3(\lambda^2 - 1)}{\lambda^2 + 1} x^2 - 3 \leq 0 \]

### Step 3: Rearranging the Inequality

Rearranging gives:

\[ \frac{3(\lambda^2 - 1)}{\lambda^2 + 1} x^2 \leq 3 \]

Dividing both sides by 3 (which is positive), we have:

\[ \frac{(\lambda^2 - 1)}{\lambda^2 + 1} x^2 \leq 1 \]

### Step 4: Analyze the Expression

Since \( x^2 \) is always non-negative, the inequality holds for all \( x \) if the coefficient of \( x^2 \) is non-positive:

\[ \frac{\lambda^2 - 1}{\lambda^2 + 1} \leq 0 \]

### Step 5: Solve the Inequality

The inequality \( \frac{\lambda^2 - 1}{\lambda^2 + 1} \leq 0 \) implies:

1. The numerator \( \lambda^2 - 1 \leq 0 \)
2. The denominator \( \lambda^2 + 1 > 0 \) (which is always true for all \( \lambda \))

Thus, we need to solve:

\[ \lambda^2 - 1 \leq 0 \]

Factoring gives:

\[ (\lambda - 1)(\lambda + 1) \leq 0 \]

### Step 6: Determine the Intervals

The critical points are \( \lambda = -1 \) and \( \lambda = 1 \). We analyze the sign of the expression in the intervals:

- For \( \lambda < -1 \): Both factors are negative, so the product is positive.
- For \( -1 < \lambda < 1 \): One factor is negative and the other is positive, so the product is negative.
- For \( \lambda > 1 \): Both factors are positive, so the product is positive.

Thus, the inequality is satisfied in the interval:

\[ [-1, 1] \]

### Conclusion

The function \( f(x) \) is a decreasing function of \( \lambda \) for:

\[ \lambda \in [-1, 1] \]