Question

If $f\left(x\right)=\{\begin{array}{cc}{e}^x+ax& x<0\\ b(x-1{)}^2& x\ge 0\end{array}$ is differentiable at $x=0$, then $(a,b)$ is

• A. $(-3,-1)$
• B. $(-3,1)$
• C. $(3,1)$
• D. $(3,-1)$

Answer 1

The correct option is B $(-3,1)$

Given that $f\left(x\right)$ is differentiable at $x=0$ , which implies $f\left(x\right)$ is continuous at $x=0$

$\Rightarrow b=1$

By using differentiable condition , we get $e^0+a=2b(0-1)$

$\Rightarrow 1+a=-2b=-2$

$\Rightarrow a=-3$

Therefore the correct option is $B$