Q105 / 180 · KCET 2025 · Mathematics
The function $f(x)=\left\{\begin{array}{ll}e^x+a x & , x \lt 0 \\ b(x-1)^2 & , \quad x \geq 0\end{array}\right.$ is differentiable at $x=0$. Then
- A$a=1, b=1$
- B$a=3, b=1$
- Correct answerC$a=-3, b=1$
- D$a=3, b=-1$
Solution
$f(x)=\left\{\begin{array}{cc}e^x+a x & , \quad x \lt 0 \\ b(x-1)^2 & , \quad x \geq 0\end{array}\right.$
$\begin{aligned} & \text { Continuity } \mathrm{LHL}=1 \\ & \qquad \mathrm{RHL}=\mathrm{b} \Rightarrow \mathrm{b}=1\end{aligned}$
Differentiability LHD $=1+\mathrm{a}$
$\begin{aligned} \text { RHD } & =-2 b \\ 1+a & =-2 b \\ a & =-3\end{aligned}$
Hence, the correct answer is option: 'C'.