Question

$f\left(x\right)=\{\begin{array}{cc}{x}^2& forx\ge 1\\ ax+b& forx<1\end{array}$

If $f$ is a differentiable function, then

• A. $a=-1,b=2$
• B. $a=2,b=-1$
• C. $a=-\frac{1}{2},b=3$
• D. None of these

Answer 1

Answer: (B)

$a=2,b=-1$

For f to be differentiable it has to be continuous
$\Rightarrow$ LHS $=$RHS $=f\left(1\right)$
$\Rightarrow a+b=1\Rightarrow \left(1\right)$
Also for $f$ to be differentiable at $x=1$, we have

LHD $=$ RHD
$\Rightarrow (2x{)}_{x=1}=a\Rightarrow a=2$
Now using (1) $b=-1$