Let `f(x)=(x^(4)-lambdax^(3)-3x^(2)+3lambdax)/(x-lambda).` If range of f(x) is the set of entire real numbers, the true set in which `lambda` lies is

Options

A. (a)[-2,2]
B. (b)[0,4]
C. (c)(1,3)
D. (d)None of these

Answer

To solve the problem, we start with the function given:

\[ f(x) = \frac{x^4 - \lambda x^3 - 3x^2 + 3\lambda x}{x - \lambda} \]

### Step 1: Simplify the function

We can simplify the function \( f(x) \) by factoring the numerator. We notice that we can group terms to factor out \( (x - \lambda) \):

\[ f(x) = \frac{x^4 - \lambda x^3 - 3x^2 + 3\lambda x}{x - \lambda} \]

### Step 2: Factor the numerator

We can rewrite the numerator:

\[ x^4 - \lambda x^3 - 3x^2 + 3\lambda x = x^3(x - \lambda) - 3x(x - \lambda) \]

This allows us to factor out \( (x - \lambda) \):

\[ = (x^3 - 3x)(x - \lambda) \]

### Step 3: Cancel the common factor

Now we can cancel \( (x - \lambda) \) from the numerator and denominator, assuming \( x \neq \lambda \):

\[ f(x) = x^3 - 3x \]

### Step 4: Analyze the range of \( f(x) \)

The function \( f(x) = x^3 - 3x \) is a polynomial function. To determine the range of \( f(x) \), we can find its critical points by taking the derivative and setting it to zero:

\[ f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x - 1)(x + 1) \]

Setting \( f'(x) = 0 \) gives us the critical points:

\[ x = 1 \quad \text{and} \quad x = -1 \]

### Step 5: Evaluate \( f(x) \) at critical points

Now we evaluate \( f(x) \) at the critical points:

\[ f(1) = 1^3 - 3(1) = 1 - 3 = -2 \]
\[ f(-1) = (-1)^3 - 3(-1) = -1 + 3 = 2 \]

### Step 6: Determine the behavior of \( f(x) \)

As \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \). Since \( f(x) \) is a cubic polynomial, it will take all real values between its local minimum and maximum.

### Step 7: Conclusion about the range

Since \( f(x) \) has a local minimum of -2 and a local maximum of 2, the range of \( f(x) \) is \( (-\infty, \infty) \). Therefore, for the range of \( f(x) \) to be the entire set of real numbers, \( \lambda \) can be any real number.

Thus, the true set in which \( \lambda \) lies is:

\[ \lambda \in \mathbb{R} \]