Question

The least positive value of $x$, satisfying $\tan x=x+1,$ lies in the interval $(\pi /4,\pi /2)$. If this is true enter 1, else enter 0.

Answer 1

Let $f\left(x\right)=\tan x$ and $g\left(x\right)=x+1$
Clearly for $x=\pi /4,f\left(x\right)=1,g\left(x\right)=1+\pi /4=>f\left(x\right)<g\left(x\right)$
and for $x=\pi /2,f\left(x\right)=\infty ,g\left(x\right)=1+\pi /2=>f\left(x\right)>g\left(x\right)$
Hence root will lie in the interval $(\pi /4,\pi /2)$
It can be easily done by plotting graph of f an g.