Question

Let $f\left(x\right)=\{\begin{array}{cc}{e}^{x^2+x}& ,x<0\\ ax+b& x\ge 0\end{array}$. Then $f\left(x\right)$ is continuous and has a derivative at $x=0,$ for $(a,b)$ equal to:

• A. $(2,1)$
• B. $(1,2)$
• C. $(1,1)$
• D. $(2,2)$

Answer 1

The correct option is C $(1,1)$

Since $f\left(x\right)$ is continuous at $0$ then

$\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}f\left(x\right)$

or, $1=b$.

Also, $f$ is differentiable at $0$ then

$L{f}^{\prime }\left(0\right)=R{f}^{\prime }\left(0\right)$

or, $e^{x^2+x}.(2x+1){|}_{x=0}=a$

or, $1=a$.

$\therefore (a,b)=(1,1)$