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Let \mathrm{f: R \rightarrow R } be a function defined as
\mathrm{f(x)=\left\{\begin{array}{ccc}5, & \text { if } & x \leq 1 \\ a+b x, & \text { if } & 1<x<3 \\ b+5 x, & \text { if } & 3 \leq x<5 \\ 35, & \text { if } & x \geq 5\end{array}\right..}Then f is

 

 

 

Option: 1

 continuous if a=-5 and b=10
 


Option: 2

 continuous if a=5 and b=5
 


Option: 3

 continuous if a=0 and b=5
 


Option: 4

 not continuous for any values of a and b


Answers (1)

 Given,\mathrm{ f(x)=\left\{\begin{array}{ccc}5, & \text { if } & x \leq 1 \\ a+b x, & \text { if } & 1<x<3 \\ b+5 x, & \text { if } & 3 \leq x<5 \\ 30, & \text { if } & x \geq 5\end{array}\right. }

Now, \mathrm{f(1)=5, f\left(1^{-}\right)=5, f\left(1^{+}\right)=a+b }.............................(i)
Also,\mathrm{ f(3)=b+15, f\left(3^{-}\right)=a+3 b, f\left(3^{+}\right)=b+15 }..................(ii)
Also, \mathrm{f(5)=35, f\left(5^{-}\right)=b+25, f\left(5^{+}\right)=35 }..................................(iii)
From (i), (ii) and (iii), clearly, f is continuous for a=-5, b=10.

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Kshitij

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