Question

Let a function $f\left(x\right)=x\ln x$ and $f\left(x\right)=0$ has atleast one real solution, then which of the following(s) is(are) correct

• A. $x\in (2,3)$
• B. $x\in (\frac{1}{2},3)$
• C. $x\in (3,4)$
• D. $x\in (\frac{1}{3},4)$

Answer 1

Answer: (B), (D)

$x\in (\frac{1}{3},4)$

Gievn : $f\left(x\right)=x\ln x$ is continuous for $x>0$
So, $I.V.T.$ is applicable for all $x>0$
Now, $\ln x>0,x>1$ and $\ln x<0,0<x<1$
$\Rightarrow f\left(x\right)=0$ has atleast one solution in $(\frac{1}{2},3)$ and $(\frac{1}{3},4)$
but no solution in $(2,3)$ and $(3,4)$