If the derivative of an odd cubic polynomial vanishes at two different values of ‘x’ then

Options

A. coefficient of `x^(3)` & x in the polynomial must be same in sign
B. coefficient of `x^(3)` & x in the polynomial must be different in sign
C. the values of ‘x’ where derivative vanishes are closer to origin as compared to the respective roots on either side of origin
D. the values of ‘x’ where derivative vanishes are far from origin as compared to the respective roots on either side of origin

Answer

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:
\[ f(x) = ax^3 + bx^2 + cx + d \]
For the polynomial to be odd, it must satisfy the condition:
\[ f(-x) = -f(x) \]
This implies that the coefficients of the even powers of \(x\) must be zero. Therefore, we set \(b = 0\) and \(d = 0\). Thus, the polynomial simplifies to:
\[ f(x) = ax^3 + cx \]

2. **Finding the Derivative**:
We find the derivative of \(f(x)\):
\[ f'(x) = \frac{d}{dx}(ax^3 + cx) = 3ax^2 + c \]

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

4. **Analyzing the Conditions**:
For \(x^2\) to have two distinct real solutions, \(-\frac{c}{3a}\) must be positive. This implies:
\[ -c > 0 \quad \text{and} \quad 3a > 0 \quad \text{or} \quad -c < 0 \quad \text{and} \quad 3a < 0 \]
Therefore, \(c\) and \(a\) must have opposite signs.

5. **Conclusion**:
Since the polynomial is odd and the derivative vanishes at two different values of \(x\), we conclude that the roots of the derivative are symmetric about the origin. Therefore, the values of \(x\) where the derivative vanishes are closer to the origin than the roots of the polynomial itself.

### Final Answer:
If the derivative of an odd cubic polynomial vanishes at two different values of \(x\), then the values of \(x\) where the derivative vanishes are closer to the origin compared to the roots of the polynomial.