Question

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

(A) the coefficient of x3 and x in the polynomial must be the same in sign

(B) the coefficient of x3 and x in the polynomial must be the different in sign

(C) the values of ‘x’ where derivative vanishes are closer to the origin as compared to the respective roots on the either side of origin.

(D) the values of ‘x’ where derivative vanishes are far from the origin as compared to the respective roots on the either side of the origin.

Answer 1

Answer is (B), (C)

See Fig.

f(x) = ax3 + bx2 + cx + d

Now, f(x) is odd. Therefore,

f(−x) = − f(x)

⇒ −ax3 − bx2 − cx − d = −ax3 + bx2 − cx + d

It gives b = 0 = d

f(x) = ax3 + cx = x (ax2 + c)

Therefore,

f′(x) = 3ax2 + c = 0

Only when x2 = − 3 c a is positive.

Therefore, c and a are of different signs.

Let − c/a = k.