Find-the-minimum-value-of-the-function-F-x-log-e-x-x-for-x-gt-0-Hence-show-that-log-e-x-x-1-For-all-x-gt-0-

Question Number 56356 by Tawa1 last updated on 15/Mar/19

Find the minimum value of the function F (x) = \log_{e} x - x for x > 0 .

Hence show that : \log_{e} x ⩽ x - 1 . For all x > 0

Commented by Tawa1 last updated on 15/Mar/19

Answered by kaivan.ahmadi last updated on 15/Mar/19

f (x) = l n x - x \Rightarrow f^{'} (x) = \frac{1}{x} - 1 = 0 \Rightarrow x = 1

i f x < 1 \Rightarrow \frac{1}{x} > 1 \Rightarrow \frac{1}{x} - 1 > 0 \Rightarrow f^{'} (x) > 0

s o m i l l a r l y i f x > 1 \Rightarrow f^{'} (x) < 0

\Rightarrow f (1) = - 1 i s m a x i m u m \Rightarrow

f (x) ⩽ - 1 \Rightarrow l n x - x ⩽ - 1 \Rightarrow

l n x ⩽ x - 1

Commented by Tawa1 last updated on 15/Mar/19

God bless you sir

Answered by tanmay.chaudhury50@gmail.com last updated on 15/Mar/19

\frac{d f (x)}{d x} = \frac{1}{x} - 1

\frac{d^{2} f}{{d x}^{2}} = \frac{- 1}{x^{2}}

f o r m i n / m a x \frac{d f}{d x} = 0

s o \frac{1}{x} - 1 = 0

x = 1

a t x = 1 \frac{d^{2} f}{{d x}^{2}} = \frac{- 1}{1^{2}} < 0

s o n o m i n i m u m v a l u e \dots

f o r m i n \frac{d^{2} f}{{d x}^{2}} > 0 a n d f o r m a x \frac{d^{2} f}{{d x}^{2}} < 0

s o a t x = 1 f (x) =∣ l n x - x ∣_{x = 1} \to - 1

v a l u e = - 1

Commented by tanmay.chaudhury50@gmail.com last updated on 15/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 15/Mar/19

f r o m g r a p h o f l n x - x i t i s c l e a r t h a t f (x) h s s m a x v a l u e a t x = 1 a n d t h a t v a l u e i s - 1

Commented by Tawa1 last updated on 15/Mar/19

God bless you sir . i appreciate your effort

Answered by tanmay.chaudhury50@gmail.com last updated on 15/Mar/19

i f \frac{d g}{d x} < \frac{d h}{d x}

t h e n g (x) < h (x)

n o w \frac{d}{d x} (l n x) = \frac{1}{x}

\frac{d}{d x} (x - 1) = 1

n o w \frac{1}{x} - 1

\frac{1 - x}{x} < 0 w h e n x > 1

s o l n x < (x - 1) w h e n x > 1

a t x = 1 i, e l n (1) = 0

(1 - 1) = 0

{(l n x)}_{x = 1} = {(x - 1)}_{x = 1}

w h e n 1 > x > 0

l n x = - v e (x - 1) = - v e

a n d ∣ l n x ∣>∣ (x - 1) ∣

s o l n x < (x - 1) w h e n 1 > x > 0

s o l n x < (x - 1) w h e n x > 0

Commented by tanmay.chaudhury50@gmail.com last updated on 15/Mar/19

Commented by Tawa1 last updated on 15/Mar/19

God bless you sir

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