Question

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

• A. has no root in (1, 3)
• 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]

Answer 1

Answer: (B)

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