Assertion (A) : if `x^4 -x^3 -6x^2 +4x +8=0 ` has a multiple root then the equation having the same root is ` 4x^3 -3x^2 - 12 x +4=0`
Reason (R ) : If ` alpha ` is repeated root of ` f(x) =0` then ` alpha `is also a root of ` f ^1 (x ) =0`

Options

A. both A and R are true R is the correct explanation of A
B. both A and R are true and R is not correct explanation of A
C. A is true and R is false
D. A is false and R is true

Answer

To solve the problem, we need to analyze the assertion and reason provided in the question step by step.

### Step 1: Define the function
We start with the polynomial given in the assertion:
\[ f(x) = x^4 - x^3 - 6x^2 + 4x + 8 \]

### Step 2: Differentiate the function
Next, we find the derivative of this function:
\[ f'(x) = \frac{d}{dx}(x^4) - \frac{d}{dx}(x^3) - \frac{d}{dx}(6x^2) + \frac{d}{dx}(4x) + \frac{d}{dx}(8) \]
Calculating each term:
- The derivative of \(x^4\) is \(4x^3\)
- The derivative of \(x^3\) is \(3x^2\)
- The derivative of \(6x^2\) is \(12x\)
- The derivative of \(4x\) is \(4\)
- The derivative of a constant (8) is 0

Thus, we have:
\[ f'(x) = 4x^3 - 3x^2 - 12x + 4 \]

### Step 3: Check for roots
Now, we need to check if \(f(x)\) has a multiple root. A multiple root \( \alpha \) means that \( f(\alpha) = 0 \) and \( f'(\alpha) = 0 \).

Let's test \( x = 2 \):
1. Calculate \( f(2) \):
\[ f(2) = 2^4 - 2^3 - 6(2^2) + 4(2) + 8 \]
\[ = 16 - 8 - 24 + 8 + 8 = 0 \]
So, \( x = 2 \) is a root of \( f(x) \).

2. Now calculate \( f'(2) \):
\[ f'(2) = 4(2^3) - 3(2^2) - 12(2) + 4 \]
\[ = 4(8) - 3(4) - 24 + 4 = 32 - 12 - 24 + 4 = 0 \]
So, \( x = 2 \) is also a root of \( f'(x) \).

### Step 4: Conclusion about the roots
Since \( x = 2 \) is a root of both \( f(x) \) and \( f'(x) \), it is a multiple root of \( f(x) \).

### Step 5: Verify the assertion
The assertion states that if \( f(x) \) has a multiple root, then the equation \( 4x^3 - 3x^2 - 12x + 4 = 0 \) has the same root. Since we have shown that \( x = 2 \) is a root of \( f'(x) \), the assertion is indeed true.

### Step 6: Verify the reason
The reason states that if \( \alpha \) is a repeated root of \( f(x) = 0 \), then \( \alpha \) is also a root of \( f'(x) = 0 \). This is exactly what we have shown through our calculations.

### Final Conclusion
Both the assertion and the reason are true, and the reason correctly explains the assertion.

### Summary
- Assertion (A) is true.
- Reason (R) is true and explains (A).