If the derivative of an odd cubic polynomial vanishes at two different values of ‘x’ then (A) coefficint of `x^(3)` & `x` in the polynomial must be of same sign (B) coefficient of `x^(3)` & `x` in the polynomial must be of different sign (C) the value of `'x'` where derivative vanishes are of same sign (D) the values of `'x'` where derivative vanishes are both positive

Options

A. coefficint of `x^(3)` & `x` in the polynomial must be of same sign
B. coefficient of `x^(3)` & `x` in the polynomial must be of different sign
C. the value of `'x'` where derivative vanishes are of same sign
D. the values of `'x'` where derivative vanishes are both positive

Explanation

To solve the problem, we need to analyze the properties of an odd cubic polynomial and its derivative.

### Step-by-Step Solution:

1. **Understanding the Odd Cubic Polynomial**:
An odd cubic polynomial can be expressed in the form:
\[ f(x) = ax^3 + bx^2 + cx + d \]
However, since it is an odd function, we have:
\[ f(-x) = -f(x) \]
This implies that the coefficients of the even powers (like \(b\) and \(d\)) must be zero. Therefore, the polynomial simplifies to:
\[ f(x) = ax^3 + cx \]

2. **Finding the Derivative**:
The derivative of the polynomial is:
\[ f'(x) = 3ax^2 + c \]

3. **Setting the Derivative to Zero**:
We are given that the derivative vanishes at two different values of \(x\). Setting the derivative to zero gives:
\[ 3ax^2 + c = 0 \]
Rearranging this, we find:
\[ x^2 = -\frac{c}{3a} \]

4. **Analyzing the Condition for Real Roots**:
For \(x^2\) to have real solutions (two different values), the right side must be positive:
\[ -\frac{c}{3a} > 0 \]
This implies that \(c\) and \(a\) must have opposite signs:
- If \(a > 0\), then \(c < 0\)
- If \(a < 0\), then \(c > 0\)

5. **Evaluating the Options**:
- **Option (A)**: Coefficient of \(x^3\) and \(x\) in the polynomial must be of the same sign. **(Incorrect)**
- **Option (B)**: Coefficient of \(x^3\) and \(x\) in the polynomial must be of different signs. **(Correct)**
- **Option (C)**: The values of \(x\) where the derivative vanishes are of the same sign. **(Incorrect)**
- **Option (D)**: The values of \(x\) where the derivative vanishes are both positive. **(Incorrect)**

### Conclusion:
The correct answer is **(B)**: The coefficient of \(x^3\) and \(x\) in the polynomial must be of different signs.