Applications Of Derivatives Question 35
Question: Maximum value of $ {{( \frac{1}{x} )}^{x}} $ is
[DCE 1999; Karnataka CET 1999; UPSEAT 2003]
Options:
A) $ {{(e)}^{e}} $
B) $ {{(e)}^{e}} $
C) $ {{(e)}^{-e}} $
D) $ {{( \frac{1}{e} )}^{e}} $
Show Answer
Answer:
Correct Answer: B
Solution:
$ f(x)={{( \frac{1}{x} )}^{x}} $
therefore $ f’(x)={{( \frac{1}{x} )}^{x}}( \log \frac{1}{x}-1 ) $
$ f’(x)=0\Rightarrow \log \frac{1}{x}=1=\log e\Rightarrow \frac{1}{x}=e\Rightarrow x=\frac{1}{e} $ Therefore maximum value of function is $ {e^{1/e}} $ .
BETA
