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

Ans: (B)
Hint : ut
$$
\begin{array}{ll}
f(x)=x \log x+x-3 & f(x)=\log x+2>0 \\
f(1)=-2, f(3)=3 \log 3, & f(1) \cdot f(3)<0
\end{array}
$$
Hence Exactly one root in \(x \in(1,3)\) as \(f(x)>0\)