Applications Of Derivatives Question 267
Question: The function $ f(x)={x^{1/x}} $ is
[AMU 2002]
Options:
A) Increasing in $ (1,\infty ) $
B) Decreasing in $ (1,\infty ) $
C) Increasing in $ (1,,e), $ decreasing in $ (e,\infty ) $
D) Decreasing in $ (1,,e), $ increasing in $ (e,\infty ) $
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Answer:
Correct Answer: C
Solution:
Let $ y={x^{1/x}} $
therefore $ \log y=\frac{1}{x}\log x $
therefore $ \frac{1}{y}\frac{dy}{dx}=\frac{1}{x^{2}}-\frac{\log x}{x^{2}}=\frac{1-\log x}{x^{2}} $
therefore $ \frac{dy}{dx}={x^{1/x}}( \frac{1-\log x}{x^{2}} ) $ Now, $ {x^{1/x}}>0 $ for all x and $ \frac{1-\log x}{x^{2}}>0 $ in (1, e) and $ \frac{1-\log x}{x^{2}}<0 $ in $ (e,\infty ) $ \ $ f(x) $ is increasing in (1, e) and decreasing in $ (e,,\infty ). $
BETA
