The equation $x log x = 2 - x$ is satisfied by at least one value of $x$ lying between $1$ and $2$. The function $f(x) = x log x$ 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 $[1,2]$

Question

Given: $5f (x) 4f(\frac 1x) = x^2 - 4 $ & $y = 9f(x) × x^2$ If y is strictly increasing, then find interval of $x$.

Answer 1

To solve the given problem, let's analyze Statements 1 and 2:

Statement-1: The equation $x \log x = 2 - x$ is satisfied by at least one value of $x$ lying between $1$ and $2$.
- The function $f(x) = x \log x$ is an increasing function in the interval $[1, 2]$ because its derivative $f'(x) = \log x + 1$ is positive for $x > 0$.
- The function $g(x) = 2 - x$ is a decreasing function because its derivative $g'(x) = -1$ is negative.
- Since one function is increasing and the other is decreasing, their graphs must intersect at some point within the interval $[1, 2]$. Hence, Statement-1 is true.
Statement-2: The function $f(x) = x \log x$ is an increasing function and $g(x) = 2 - x$ is a decreasing function.
- This statement illustrates the behavior of both functions.
- The increasing/decreasing nature of these functions explains why they must intersect, supporting the claim of Statement-1, making Statement-2 true and correctly explaining Statement-1.

The correct option is that both Statement-1 and Statement-2 are true, and Statement-2 is a correct explanation for Statement-1.

The Correct Option is A