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  • If   <img alt="\mathrm{f(x)=\left\{\begin{array}{lc}a x^2+b, & b \neq 0, x \leq 1 \\ b x^2+a x+c, & x>1\end{array}\right.}" src="https://entrancecorner.oncodecogs.com/gif.latex

If   \mathrm{f(x)=\left\{\begin{array}{lc}a x^2+b, & b \neq 0, x \leq 1 \\ b x^2+a x+c, & x>1\end{array}\right.} 
Then f(x) is continuous and differentiable at x=1 if :

Option: 1

\mathrm{c=0, a=2 b}


Option: 2

\mathrm{a=b, c \in R}


Option: 3

\mathrm{a=b, c=0}


Option: 4

\mathrm{a=b, c \neq 0}


Answers (1)

best_answer

Since f(x) is continuous at x=1, therefore
\mathrm{ \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x) \Rightarrow a+b=b+a+c \Rightarrow c=0 }
As f(x) is differentiable at x=1, therefore

\mathrm{ \text { (LHD at } x=1)=(\text { RHD at } x=1) \\ }
\mathrm{ \Rightarrow \quad 2 a+0=2 b+a+0 \Rightarrow a=2 b . \\ }

\mathrm{ \text { Hence, } a=2 b, c=0 . }

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