Limit Continuity and Differentiability 7 Question 24
24. There exists a function $f(x)$ satisfying $f(0)=1$, $f^{\prime}(0)=-1, f(x)>0, \forall x$ and
$(1982,2 M)$
(a) $f^{\prime \prime}(x)<0, \forall x$
(b) $-1<f^{\prime \prime}(x)<0, \forall x$
(c) $-2 \leq f^{\prime \prime}(x) \leq-1, \forall x$
(d) $f^{\prime \prime}(x)<-2, \forall x$
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Answer:
Correct Answer: 24. $(b, c)$
Solution:
- Let $u=\sec ^{-1}-\frac{1}{2 x^{2}-1}$ and $v=\sqrt{1-x^{2}}$
Put $\quad x=\cos \theta$
$\therefore \quad u=\sec ^{-1}(-\sec 2 \theta)$ and $v=\sin \theta$
$\Rightarrow \quad u=\pi-2 \theta \quad\left[\because \sec ^{-1}(-x)=\pi-\sec ^{-1} x\right]$
and $\quad v=\sin \theta$
$\Rightarrow \quad \frac{d u}{d \theta}=-2$
an $\quad d \frac{d v}{d \theta}=\cos \theta$
$\Rightarrow \quad \frac{d u}{d v}=-\frac{2}{\cos \theta}, \quad \frac{d u}{d v}{ } _{\theta=\pi / 3}=-4$
