Question

Statement -1 : The equation $x \ln x=2-x$ is satisfied by atleast one value of $x$ lying between 1 and 2 .
Statement-2: The function $f ( x )= x \ln x$ is an increasing function in $[1,2]$ and $g ( x )=2- x$ is a decreasing function in $[1,2]$ and the graphs represented by these functions intersect at a point in $[1,2]$.

• A. Statement-1 is true, statement- 2 is true and statement- 2 is correct explanation for statement- 1 .
• B. Statement- 1 is true, statement- 2 is true and statement- 2 is NOT the correct explanation for statement- 1 .
• C. Statement- 1 is true, statement- 2 is false.
• D. Statement-1 is false, statement- 2 is true.

Answer 1

Answer: (A)

Statement-1: Let $F ( x )= x \ln x + x -2$, which is a continuous function in $[1,2]$.
Also, $F (1)=-1<0$ and $F (2)=2 \ln 2>0 \Rightarrow F (1) \cdot F (2)<0$
So, using I.V.T. , $F ( c )=0, c \in(1,2)$.
Statement -2: $f ( x )= x \ln x \Rightarrow f ^{\prime}( x )=1+\ln x >0, x \in[1,2]$ $\therefore f ( x )$ is increasing in $[1,2]$.
Also, $g ( x )=2- x$, which is decreasing in $[1,2]$.