The equation x log x = 3 - x

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]

Tardigrade Admin · Accepted Answer

Let f(x)=x log x+x-3 ⇒ f'(x)=x ⋅ (1/x)+ log x+1 ⇒ f'(x)= log x+2>0 ⇒ f(1)=-2 and f(3)=3 log 3, f(1) ⋅ f(3)<0 Hence, exactly one root in x ∈(1,3) as f(x)>0

Q. The equation $x lo g x = 3 - x$

has no root in (1, 3)

has exactly one root in (1, 3)

x log x - (3 - x) > 0 in [1, 3]

x log x - (3 - x) < 0 in [1, 3]

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

Q. The equation xlogx=3−x