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  • Let <img alt="\mathrm{f(x)=\left\{\begin{array}{cc}-3+|x|, & -\infty<x<1 \\ a+|2-x|, & 1 \leq x<\infty\end{array}\right.}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cm

Let \mathrm{f(x)=\left\{\begin{array}{cc}-3+|x|, & -\infty<x<1 \\ a+|2-x|, & 1 \leq x<\infty\end{array}\right.} and
\mathrm{g(x)=\left\{\begin{array}{cc}2-|-x| . & -\infty<x<2 \\ -b+\operatorname{sgn}(x), & 2 \leq x<\infty\end{array}\right.}

where \mathrm{\operatorname{sgn}(x)} denotes signum function of \mathrm{x}. If \mathrm{h(x)=f(x)+g(x)} is discontinuous at exactly one point, then which of the following is not possible?

Option: 1

a=-3, b=0


Option: 2

a=0, b=1


Option: 3

a=2, b=1


Option: 4

a=-3, b=1


Answers (1)

best_answer

\mathrm{h(x)=f(x)+g(x)}

\mathrm{=\left\{\begin{array}{cc} -1, & -\infty<x<1 \\ a+4-2 x, & 1 \leq x<2 \\ a-b-1+x, & 2 \leq x<\infty \end{array}\right.}

\therefore  We must have either \mathrm{a=-3, b \neq 1 \, or \, b=1, a \neq-3}

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manish painkra

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