Question

For all \[{{\log }_{e}}1-1\]\[x\in \left( 0,1 \right)\]
(a) \[{{e}^{x}}<1+x\]
(b) \[log_e(1+x)< x\]
(c) sin x > x
(d) \[{{\log }_{e}}x>x\]

Answer 1

Hint:In this question, we check all the options by taking f (x) or g (x). Then we differentiate it and check whether it is an increasing or decreasing function and find out which option holds.

Complete step by step Solution:
We have an equation with four options and we have to find out which option is correct.
First, we check option (a)
Let f (x) = \[{{e}^{x}}-1-x\]
Then \[f'(x)={{e}^{x}}-1\] for \[x\in \left( 0,1 \right)\]
\[{{e}^{x}}-1>0\] for \[x\in \left( 0,1 \right)\]
F’ (x) > 0
As f (x) is an increasing function.
F (x) > F (0) , \[\forall \]\[x\in \left( 0,1 \right)\]
F (0) = \[{{e}^{0}}\] -1 -0 = 0
\[{{e}^{x}}-1-x\] > 0
\[{{e}^{x}}>1+x\]
Option (a) does not hold.
Now we check the (b) part
Let g (x) = Log (1+x) – x
By differentiating both sides w.r.t x, we get
Then \[g'(x)=\frac{1}{1+x}-1\]
Which is equal to \[g'(x)=\frac{-x}{1+x}\] < 0, \[\forall \]\[x\in \left( 0,1 \right)\]
G (x) is decreasing on (0,1)
Therefore, x > 0
G (x) < g (0)
G(0) = \[{{\log }_{e}}(1+0)-0\]
         = \[{{\log }_{e}}1-0\] = 0
\[{{\log }_{e}}(1+x)-x<0\]
\[log_e(1+x)< x\]
Option (b) is correct.
Now we check option (c)
Let h (x) = sin x – x
Now we differentiate both sides w.r.t x, we get
H’(x) = cos x -1 for \[x\in \left( 0,1 \right)\]
H’ (x) < 0
H’(x) is a decreasing function.
Then h (x) < h (0)
H (0) = sin 0 – 0
          = 0
Sin x -x < 0
Sin x < x
Option (c) does not hold
Now we check option (d)
Let P (x) = \[{{\log }_{e}}x-x\]
Now we differentiate both sides of equation w.r.t x, we get
P’(x) = \[\frac{1}{x}-1\] for \[x\in \left( 0,1 \right)\]
P’(x) > 0
This means p (x) is an increasing function.
P(1) > P(x) > 0 ---------------- (1)
P(0) = \[{{\log }_{e}}0-0\] = \[-\infty \]
P(1) = \[{{\log }_{e}}1-1\] = -1
By putting these values in equation (1), we get
-1 > \[{{\log }_{e}}x-x\] > \[-\infty \]
\[{{\log }_{e}}x-x\]< 0
\[log_eX>X\]
Option (d) is not correct.

Hence, the correct option is (b).

Note: in these types of questions, students made mistakes in finding out whether it is an increasing function or decreasing function. We should take care of that type of minor mistake.