If `f(x)` is a cubic polynomial such that `f(x)=-2/x` for x=2,3,4 and 5 then the value of `52-f(10)` is

Answer

To solve the problem, we need to find the value of \( 52 - f(10) \) given that \( f(x) \) is a cubic polynomial such that \( f(2) = f(3) = f(4) = f(5) = -\frac{2}{x} \).

### Step-by-Step Solution:

1. **Understanding the Problem**: We know that \( f(x) \) is a cubic polynomial and that it equals \( -\frac{2}{x} \) at \( x = 2, 3, 4, 5 \). This means that \( f(x) + \frac{2}{x} = 0 \) at these points.

2. **Setting Up the Equation**: We can rewrite the condition as:
\[ x f(x) + 2 = 0 \quad \text{for } x = 2, 3, 4, 5. \]
This implies:
\[ x f(x) = -2. \]

3. **Formulating the Polynomial**: Since \( f(x) \) is a cubic polynomial, we can express it as:
\[ x f(x) + 2 = k(x - 2)(x - 3)(x - 4)(x - 5), \]
where \( k \) is a constant.

4. **Finding the Constant \( k \)**: To find \( k \), we can substitute \( x = 0 \):
\[ 0 \cdot f(0) + 2 = k(-2)(-3)(-4)(-5). \]
This simplifies to:
\[ 2 = k \cdot 120 \implies k = \frac{2}{120} = \frac{1}{60}. \]

5. **Substituting Back**: Now we have:
\[ x f(x) + 2 = \frac{1}{60}(x - 2)(x - 3)(x - 4)(x - 5). \]

6. **Finding \( f(10) \)**: We substitute \( x = 10 \) into the equation:
\[ 10 f(10) + 2 = \frac{1}{60}(10 - 2)(10 - 3)(10 - 4)(10 - 5). \]
This becomes:
\[ 10 f(10) + 2 = \frac{1}{60}(8 \cdot 7 \cdot 6 \cdot 5). \]
Calculating the right side:
\[ 8 \cdot 7 = 56, \quad 56 \cdot 6 = 336, \quad 336 \cdot 5 = 1680. \]
Thus:
\[ 10 f(10) + 2 = \frac{1680}{60} = 28. \]

7. **Solving for \( f(10) \)**:
\[ 10 f(10) = 28 - 2 = 26 \implies f(10) = \frac{26}{10} = 2.6. \]

8. **Calculating \( 52 - f(10) \)**:
\[ 52 - f(10) = 52 - 2.6 = 49.4. \]

### Final Answer:
The value of \( 52 - f(10) \) is \( 49.4 \).