Question

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

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

Answer 1

Answer: (C)

$f^{\prime }(x)\ge f^3(x)+\frac{1}{f(x)}$ or
$f^{\prime }(x)\cdot f(x)\ge 1+f^4(x)$
or $\frac{f(x)\cdot f^{\prime }(x)}{1+f^4(x)}\ge 1$
Integrating with respect to $x$,
from $x=a$ to $x=b$
$\frac{1}{2}{\left({\tan }^{-1}\left(f^2(x)\right)\right)}_a^b\ge b-a$
or $(b-a)\le \frac{1}{2}\{\underset{x\to b^{-}}{\lim }\left({\tan }^{-1}\left(f^2(x)\right)\right\}$
$-\underset{x\to a^{+}}{\lim }\left({\tan }^{-1}\left(f^2(x)\right)\right)$
$\Rightarrow b-a\le \frac{\pi }{24}$