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If `f(x)={{:(,e^(x),x lt 2),(,ax+b,x ge 2):}` is differentiable for all `x in R`, themA. `a=e^(2),b=-e^(2)`B. `a=-e^(2),b=e^(2)`C. `a=b=e^(2)`D. none of these

Answer» Correct Answer - A
Clearly, f(x) is everywhere continuous and differentiable except possible at x=2.
At x=2, we have
`underset(x to 2^(-))lim f(x)=underset(x to 2^(-))lim e^(x)=e^(2)`
`underset(x to 2^(+))lim f(x)=underset(x to 2^(+))lim ax+b=2a+b`
`f(2)=2a+b` Also,
`("LHD at x=2")=((d)/(dx)(e^(x)))_(x=2)=e^(2)`
`("RHD at x=2")=((d)/(dx)(ax+b))_(x=2)=a`
For f(x) to be differentiable at x=2, it should be both continuous and differentiable at x=2
`therefore underset(x to 2^(-))lim f(x)=underset(x to 2^(+))lim f(x)=f(2)`
and (LHD at x=2)=(RHD at x=2)
`Rightarrow e^(2)=2a+b and a=e^(2) Rightarrow a=e^(2) and b=-e^(2)`


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