Q48 / 155 · WBJEE 2024 · Mathematics

Consider the function $f(x)=(x-2) \log _e \cdot x$. Then the equation $x \log _e x=2-x$

  • Correct answer
    A
    has at least one root in (1, 2)
  • B
    has no root in (1, 2)
  • C
    is not at all solvable
  • D
    has infinitely many roots in (–2, 1)

Solution

$\begin{aligned} & \text { Hint : Let } g(x)=x \log _e x-2+x \\ & g^{\prime}(x)=1+\log _e x+1=2+\log _e x \\ & g(1) \cdot g(2)=-1 \times\left(2 \log _e 2\right)=-v e\end{aligned}$

Hence, the correct answer is option: 'A'.

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