f ( x ) = x l o g e x , x ≠ 1 , is decreasing in interval Options: A . &nbsp ( 0 , e ) B . &nbsp ( 1 , e ) C . &nbsp ( e , ∞ ) D . &nbsp R

Answer: Option A : A f'(x) = l o g e x . 1 − x . 1 x ( l o g e x ) 2 = ( l o g e x − 1 ) ( l o g e x ) 2 < 0 It is decreasing at (0, e) – {1}

f(x) = xloge x, x ≠ 1, is decreasing in interval

Question -

$f (x) = \frac{x}{l o g_{e} x}, x \neq 1$ , is decreasing in interval

Options:

A .

(0, e)

B .

(1, e)

C .

(e, \infty)

D . R

Answer: Option A
:
A
f'(x) =

\frac{l o g_{e} x . 1 - x . \frac{1}{x}}{(l o g_{e} x)^{2}}

\frac{(l o g_{e} x - 1)}{(l o g_{e} x)^{2}} < 0

It is decreasing at (0, e) – {1}

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