Applications Of Derivatives Question 365
Question: The maximum value of $ {x^{1/x}} $ is
[MP PET 2004]
Options:
A) $ \frac{1}{e} $
B) $ {e^{1/e}} $
C) e
D) $ \frac{1}{e^{e}} $
Show Answer
Answer:
Correct Answer: B
Solution:
$ y={x^{1/x}} $ , Taking log , we have $ \log y=\frac{1}{x}\log x $
Differentiate both sides w.r.t. x $ \frac{1}{y}\frac{dy}{dx}=\frac{1}{x^{2}}-\frac{\log x}{x^{2}} $
therefore $ \frac{dy}{dx}=\frac{1}{x^{2}}(1-\log x){x^{1/x}} $
For maximum, $ \frac{dy}{dx}=0 $
therefore $ x=e $ ; \ $ {y_{\max }}={e^{1/e}} $ .
BETA
