JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let a function \(f: R \rightarrow R\) be defined as \(f(x)=\sin x-e^{x} \,\,\,\, \text { if } x \leq 0\) \(\quad\quad\quad a+[-x] \,\,\,\, \text { if } 0\,<\,x\,<\,1\) \(\quad\quad\quad 2 x-b \,\,\,\,\,\,\,\, \text { if } \geq 1\) where \([\mathrm{x}]\) is the greatest integer less than or equal to \(\mathrm{x}\). If \(\mathrm{f}\) is continuous on \(\mathrm{R}\), then \((\mathrm{a}+\mathrm{b})\) is equal to:
- A \(5\)
- B \(3\)
- C \(2\)
- D \(4\)
Answer & Solution
Correct Answer
(B) \(3\)
Step-by-step Solution
Detailed explanation
Continuous at \(x=0\) \(f\left(0^{+}\right)=f^{-} \Rightarrow a-1=0-e^{0}\) \(\Rightarrow a=0\) Continuous at \(\mathrm{x}=1\) \(f\left(1^{+}\right)=f(1^{-})\) \(\Rightarrow 2(1)-b=a+(-1)\) \(\Rightarrow b=2-a+1 \Rightarrow b=3\) \(\therefore a+b=3\)
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