Question

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

Answer 1

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.