A) \[\left( 2e-1,\text{ }2e \right)\]
B) \[(e-1,\,\,2e-1)\]
C) \[\left( \frac{e-1,}{2}\,\,,\,\,e-1 \right)\]
D) \[\left( 0\,\,,\,\,\frac{e-1}{2} \right)\]
Correct Answer: D
Solution :
| Given, \[f'(x)<2f(x)\] |
| \[\Rightarrow \]\[\frac{f'(x)}{f(x)}<2\] |
| On integrating, we get \[f(x)<C{{e}^{2x}}\] |
| Put \[x=\frac{1}{2}\]\[\Rightarrow \]\[C>\frac{1}{e}\] |
| Hence, \[f(x)<\frac{{{e}^{2x}}}{e}\] |
| \[\Rightarrow \]\[f(x)<{{e}^{2x-1}}\] |
| \[\Rightarrow \]\[0<\int\limits_{1/2}^{1}{f(x)dx<}\int\limits_{1/2}^{1}{{{e}^{2x-1}}dx}\] |
| \[0<\int\limits_{1/2}^{1}{f(x)<\frac{e-1}{2}}\] |
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