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

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

Answer

B

Explanation

Let `f(x)=ax^(3)+bx^(2)+cx+d`
`f(-x)=-ax^(2)+bx^(2)-cx+d`
since `f(x)` is odd
`:.ax^(3)+bx^(2)+cx+d=ax^(2)-bx^(2)+cx-d`
`d=0, b=0`
`:.f(x)=ax^(3)+cx`
`f^(')(x)=3ax^(2)+c=0 3ax^(2)=-c`
`x^(2)-c/(3a)gt0`
`:. a` and `c` must be of different signs.