Question

Find the values of $a$ and $b$ so that the function $f\left(x\right)=\{\begin{array}{ccc}{x}^2+3x+a,& if& x\le 1\\ bx+2,& if& x>1\end{array}$ is differentiable at each $x\in R$.

Answer 1

$f\left(x\right)=\{\begin{array}{ccc}{x}^2+3x+a,& if& x\le 1\\ bx+2,& if& x>1\end{array}$

$f\left(x\right)$ is differentiable

$\Rightarrow \underset{x\to 1^{-}}{lt}{f}^1\left(x\right)=\underset{x\to 1^{+}}{lt}{f}^1\left(x\right)$

$\Rightarrow \underset{x\to 1^{-}}{lt}(2x+3)=b$

$\therefore b=2+3=5$

$\therefore f\left(x\right)$ is continuous as it is differentiable

$\Rightarrow \underset{x\to 1^{-}}{lt}f\left(x\right)=\underset{x\to 1^{+}}{lt}f\left(x\right)$

$\Rightarrow \underset{x\to 1^{-}}{lt}(x^2+3x+a)=\underset{x\to 1^{+}}{lt}(bx+2)$

$\Rightarrow 1+3+a=\underset{x\to 1^{+}}{lt}(5x+2)$

$\therefore a=7-4=3$.