The correct option is C (−2,2)
Let f(x)=x3−3x+a
f′(x)=3x2−3=3(x−1)(x+1)
f′(x)=0⇒x=1,−1
At x=1, f′′(x)>0, hence x=1 is a point of maxima and similarly x=−1 is a point of minima.
For real roots, the maxima should have a value >0 and the minima should be <0.
Now, f(1)=a−2, f(−1)0=a+2
Hence,
a−2<0 and a+2>0
Thus, the given equation would have real and distinct roots, if a∈(−2,2).
Hence, option 'B' is correct.