Applications Of Derivatives Question 351
Question: The function $ f(x)={x^{-x}},,(x,\in ,R) $ attains a maximum value at x =
[EAMCET 2002]
Options:
A) 2
B) 3
C) 1/e
D) 1
Show Answer
Answer:
Correct Answer: C
Solution:
$ f(x)=y={x^{-x}} $
therefore $ \log y=-,x\log x $ Differentiating w.r.t. x, $ \frac{1}{y}.\frac{dy}{dx}=-[ x.\frac{1}{x}+\log x ] $
therefore $ \frac{1}{y}.\frac{dy}{dx}=-[1+\log x] $
therefore $ \frac{dy}{dx}=-{x^{-x}}[1+\log x] $
therefore $ \frac{dy}{dx}={x^{-x}}[ \log \frac{1}{x}-1 ] $ Put $ \frac{dy}{dx}=0 $
therefore $ {\log_{e}}\frac{1}{x}={\log_{e}}e $
therefore $ \frac{1}{x}=e\Rightarrow x=\frac{1}{e} $ .
BETA
