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Question

If $$f(x)=\begin{cases}e^{x}, x< 2\\  a+bx, x\geq 2 \end{cases}$$, is differentiable for all $$x\in R$$ then


A
a+b=0
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B
a+2b=e2
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C
b=e2
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D
none of these
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Solution

The correct options are
A $$a+2b=e^{2}$$
C $$b=e^{2}$$
D $$a+b=0$$
For given function to be differentiable, it has to be continuous.
$$\Rightarrow f(2-)=f(2+)=f(2) => a+2b = e^2$$
Also $$f'(2+)=f'(2-)\Rightarrow e^2=b \Rightarrow a=-e^2 $$

Mathematics

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