If `f(x)=Pi_(k=1)^(999)(x^(2)-47x+k)`. then product of all real roots of `f(x)=0` is

Options

A. `550!`
B. `551!`
C. `552!`
D. `999!`

Answer

To solve the problem, we need to find the product of all real roots of the function defined as:

\[ f(x) = \prod_{k=1}^{999} (x^2 - 47x + k) \]

### Step 1: Analyze the Quadratic Equation
The quadratic equation \(x^2 - 47x + k\) has real roots if its discriminant is non-negative. The discriminant \(D\) is given by:

\[ D = b^2 - 4ac = (-47)^2 - 4(1)(k) = 2209 - 4k \]

For the roots to be real, we need:

\[ 2209 - 4k \geq 0 \]

### Step 2: Solve the Inequality
Rearranging the inequality gives:

\[ 2209 \geq 4k \implies k \leq \frac{2209}{4} = 552.25 \]

Since \(k\) is an integer, the maximum integer value for \(k\) is 552.

### Step 3: Determine the Roots
The roots of the quadratic equation \(x^2 - 47x + k\) can be found using the quadratic formula:

\[ x = \frac{47 \pm \sqrt{D}}{2} \]

The product of the roots of the quadratic \(x^2 - 47x + k\) is given by:

\[ \text{Product of roots} = \frac{k}{1} = k \]

### Step 4: Calculate the Total Product of Roots
Since \(f(x)\) is the product of 999 such quadratic equations, we need to consider the product of all real roots for each \(k\) from 1 to 552.

For each \(k\) from 1 to 552, the product of the roots for each quadratic \(x^2 - 47x + k\) is \(k\). Therefore, the total product of all real roots is:

\[ \prod_{k=1}^{552} k = 552! \]

### Step 5: Conclusion
Thus, the product of all real roots of \(f(x) = 0\) is:

\[ \boxed{552!} \]