The maximum value of log_e ⁡xx, if x0 is

Question

The maximum value of log_e ⁡xx, if x>0 is

Accepted Answer

Correct answer: B. (b) : Let y= log_e ⁡xx ⇒ dy dx= x^ 1x- log ⁡xx^2 ⇒ dy dx= 1- log ⁡xx^2 ⇒ d^2ydx^2= x^2 ( -1x )+ (1- log ⁡x ) 2xx^4 = x-2x log ⁡xx^4 For maxima/minima, dy dx=0 ⇒1- log ⁡x=0 ⇒x=e And ( d^2ydx^2 )_ (x=e )= -ee^4<0 ∴ y is maximum at x=e. And maximum value of y= 1e.

$The maximum value of \frac{log _{e} x}{x}, if x > 0 is$

Solution

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