The correct option is A 12
We have f(x)=ax2+bx+c.
⇒f(x+h)=a(x+h)2+b(x+h)+c
Also, f′(x)=2ax+b
⇒f′(x+θh)=2a(x+θh)+c
Putting these values in Lagrange's mean value theorem we get
∴f(x+h)=f(x)+hf′(x+θh)a(x+h)2+b(x+h)+c=ax2+bx+c+h[2a(x+θh)+b]
when x→0. we have ah2+bh+c+h[2aθh+b]
ah2=2aθh2 or θ=12
which is the required value of θ.