Question

$If f(x) = \begin{cases} \text{if $ax^{2} + b, \,\,\, b \neq 0, x \leq 1$} \\[2ex] \text{if $bx^{2} +ax +c, \,\,\, x > 1$ } \end{cases}$ then
$f(x)$ is continuous and differentiable at $x=1$, if

• A. $c = 0,\,a = 2\,b $
• B. $ a=b,\,c\in R $
• C. $ a=b,\,c=0 $
• D. $ a=b,\,c\ne 0 $

Answer 1

Answer: (A)

Given, $f(x) =\begin{cases} a x^{2}+b, & b \neq 0, x \leq 1 \\ b x^{2}+a x+c, & x>1 \end{cases}.$
$\Rightarrow f'(x) =\begin{cases} 2 a x, & b \neq 0, x \leq 1 \\ 2 b x+a, & x > 1 \end{cases}$
Since, $f(x)$ is continuous at $x=1$
$\therefore \displaystyle\lim _ {x \rightarrow 1^{-}} f(x)=\displaystyle\lim _ {x \rightarrow 1^{+}} f(x)$
$\Rightarrow a+b=b+a+c$
$\Rightarrow c=0$
Also, $f(x)$ is differentiable at $x=1$.
$\therefore ( LHD$ at $x=1)=( RHD$ at $x=1$
$\Rightarrow 2 a=2 b(1)+a$
$\Rightarrow a=2 b$