Question

Lef $f( x )$ be a positive, continuous and differentiable function on the interval $(a, b)$. If $\lim _{x \rightarrow a^{+}} f(x)=1$ and $\lim _{x \rightarrow b^{-}} f(x)=3^{1 / 4}$. Also $f'(x) \geq f^{3}(x)+\frac{1}{f(x)}$ then

• A. $b-a \geq \frac{\pi}{4}$
• B. $b-a \leq \frac{\pi}{4}$
• C. $b-a \leq \frac{\pi}{24}$
• D. None of these

Answer 1

Answer: (C)

$f'(x) \geq f^{3}(x)+\frac{1}{f(x)} $ or
$f'(x) \cdot f(x) \geq 1+f^{4}(x)$
or $\frac{f(x) \cdot f'(x)}{1+f^{4}(x)} \geq 1$
Integrating with respect to $x$,
from $x=a$ to $x=b$
$\frac{1}{2}\left(\tan ^{-1}\left(f^{2}(x)\right)\right)_{a}^{b} \geq b-a $
or $(b-a) \leq \frac{1}{2}\left\{\displaystyle\lim _{x \rightarrow b^{-}}\left(\tan ^{-1}\left(f^{2}(x)\right)\right\}\right.$
$-\displaystyle\lim _{x \rightarrow a^{+}}\left(\tan ^{-1}\left(f^{2}(x)\right)\right)$
$\Rightarrow b-a \leq \frac{\pi}{24}$