Question

Find the greatest possible integral value of $\frac{b-a}{{\tan }^{-1}b-{\tan }^{-1}a}$, where $0<a<b<\sqrt{3}$

Answer 1

Answer: 0003

Let $\mathrm{f}(\mathrm{x})={\tan }^{-1}\mathrm{x},\mathrm{x}\in [\mathrm{a},\mathrm{b}]$
$\therefore$ Using LMVT, we get $\frac{{\tan }^{-1}\mathrm{b}-{\tan }^{-1}\mathrm{a}}{\mathrm{b}-\mathrm{a}}=\frac{1}{1+{\mathrm{c}}^2}$, where $0<\mathrm{a}<\mathrm{c}<\mathrm{b}<\sqrt{3}$
So, $1<\left(\frac{b-a}{{\tan }^{-1}b-{\tan }^{-1}a}\right)<4$
As, $\frac{1}{4}<\frac{1}{1+{\mathrm{c}}^2}<1$
$\Rightarrow$ The greatest possible integral value is 3.