Q104 / 155 · WBJEE 2018 · Mathematics

The equation $x \log x=3-x$

  • A
    has no root in (1,3)
  • Correct answer
    B
    has exactly one root in (1,3)
  • C
    $x \log x-(3-x)>0$ in [1,3]
  • D
    $x \log x-(3-x) < 0$ in [1,3]

Solution

Let $\quad f(x)=x \log x+x-3$ $\Rightarrow \quad f^{\prime}(x)=x \cdot \frac{1}{x}+\log x+1$ $\Rightarrow \quad f^{\prime}(x)=\log x+2>0$ $\Rightarrow f(1)=-2$ and $f(3)=3 \log 3, f(1) \cdot f(3 < 0)$ Hence, exactly one root in $x \in(1,3)$ as $f(x)>0$

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

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