The correct option is B [14,∞)
We know for a function to be an onto function, Range should be equal to co-domain.
Here co-domain which is given is - [0.π2)
To have the range [0.π2) domain should have all non zero values.
x2+x+a≥0
this quadratic will be non zero if its discriminant is less than and equal to zero, Since it has coefficient of x2 positive.
Discriminant ⇒1−4a≤0
⇒a≥14