Question
If the derivative of the function \[f\left( x \right) = \left\{ \begin{array}{l}
\,\,\,\,a{x^2} + b\,\,\,\,\,\,\,\,\,\,\,x < - 1\\
b{x^2} + ax + a\,\,\,x \ge - 1
\end{array} \right.\] is every where continuous, then what are the values of $$a$$ and $$b\,?$$
A.
$$a = 2,\,b = 3$$
B.
$$a = 3,\,b = 2$$
C.
$$a = - 2,\,b = - 3$$
D.
$$a = - 3,\,b = - 2$$
Answer :
$$a = 2,\,b = 3$$
Solution :
\[\begin{array}{l}
{\rm{Derivative\, of\, }}f\left( x \right) = \left\{ \begin{array}{l}
\,\,\,\,a{x^2} + b\,\,\,\,\,\,\,\,\,\,\,\,\,\,x < - 1\\
b{x^2} + ax + a\,\,\,x \ge - 1
\end{array} \right.{\rm{ \,is\,}}\\
f'\left( x \right) = \left\{ \begin{array}{l}
\,\,\,\,2ax\,\,\,\,\,\,\,\,\,\,\,\,\,x < - 1\\
2bx + a\,\,\,\,\,x \ge - 1
\end{array} \right.
\end{array}\]
If $$f'\left( x \right)$$ is continuous everywhere then it is also continuous at $$x = - 1$$
$$\eqalign{
& {\left. {f'\left( x \right)} \right|_{x = - 1}} = - 2a = - 2b + a \cr
& {\text{or, }}3a = 2b.....({\text{i}}) \cr} $$
From the given choice $$a = 2,\,b = 3$$ satisfied this equation.