Question

$f\left(x\right)=A+B{x}^2$ $\text{for}x>1$

$=3Ax-B{x}^2$ $\text{for}x\le 1$

Find A, B so that $f\left(x\right)$ is derivative at $x=1$

Answer 1

For $f\left(x\right)$ to be derivable at $x=1$. it should be continuous at $x=1$.

Hence,

$\underset{x\to 1^{-}}{lim}=\underset{x\to 1^{+}}{lim}$

$\Rightarrow A+B=3A-B\Rightarrow A=B$

So, for all values of $A=B$; $f\left(x\right)$ is continuous as well as derivable at $x=1$