Let a and b be real numbers such that the function
`g(x)={{:(,-3ax^(2)-2,x lt 1),(,bx+a^(2),x ge1):}` is differentiable for all `x in R`
Then the possible value(s) of a is (are)

Options

A. `1,2`
B. `3,4`
C. `5,6`
D. `8,9`

Answer

A

Explanation

It is given that f(x) is differentiable for all `x in R`. So, it has to be differentiable and hence continuous at x=1 also. Continuity of f(x) at x=1: If f(x) is continuous at x=1. Then
`underset(x to 1^(-))lim f(x)=underset(x to 1^(+))lim (bx+a^(2))=bxx1+a^(2)`
`Rightarrow -3a+2=b+a^(2)`
`Rightarrow a^(2)+3a+2+b=0...(i)`
Differenetiability of f(x) at x=1: If f(x) is differentiable at x=1, then
(LHD of f(x) at x=1)=(RHD of f(x) at x=1)
`Rightarrow {(d)/(dx)(-3ax^(2)-2)}_(x=1)={(d)/(dx)(bx+a^(2))}_(x=1`
`Rightarrow -6a=b.......(ii)`
Solving (i) and (ii), we get `a=1,b=-6or, a=2, b=-12`