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 $[1,2]$.