A) \[{{e}^{x}}<1+x\]
B) \[{{\log }_{e}}(1+x)<x\]
C) \[\sin x>x\]
D) \[{{\log }_{e}}x>x\]
Correct Answer: B
Solution :
Both \[{{e}^{x}}\] and \[1+x\] are increasing and \[\sqrt{e}\ge 1+\frac{1}{2},\] because \[\sqrt{e}=1.65\] nearly. so the answer is not correct. Since \[\sin \frac{\pi }{6}<\frac{\pi }{6}\] because \[\frac{1}{2}<\frac{22}{42}\]. So, is not correct. \[\log \frac{1}{2}<\frac{1}{2}\] because \[\log \frac{1}{2}\] is negative. \[\therefore \] Option is not correct. Thus, by elimination is correct.
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