Question

By LMVT, which of the following is true for x>1

• A. 1+x In x<x<1+In x
  
• B. 1+In x<x<1+x In x
• C. x <1 + x In x<1+In x <x
• D. 1+In x <1+x In x<x

Answer 1

The correct option is B

1+In x<x<1+x In x

Let f(x) =1+x In x in (1, x)
$\therefore f^{\prime }\left(x\right)=(1+Inx)>0(\because x>1)$
By LMVT, for 1 < c < x
$f^{\prime }\left(c\right)=\frac{f\left(x\right)-f\left(1\right)}{x-1}=\frac{(1+xInx-1)}{x-1}=\frac{xInx}{x-1}>0$
or x In x > x -1 or x In x+1 > x
Also, similarly by LMVT
We get, 1 + In x < x
Hence, 1+In x < x < 1 + x In x.