Limit Continuity and Differentiability 7 Question 12
12. If $f$ and $g$ are differentiable functions in $(0,1)$ satisfying $f(0)=2=g(1), \quad g(0)=0$ and $f(1)=6$, then for some $c \in 0,1[$
(2014 Main)
(a) $2 f^{\prime}(c)=g^{\prime}(c)$
(b) $2 f^{\prime}(c)=3 g^{\prime}(c)$
(c) $f^{\prime}(c)=g^{\prime}(c)$
(d) $f^{\prime}(c)=2 g^{\prime}(c)$
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Answer:
Correct Answer: 12. $(a, b, d)$
Solution:
- Given, $y=\sec \left(\tan ^{-1} x\right)$
$$ \begin{aligned} \text { Let } & & \tan ^{-1} x & =\theta \\ \Rightarrow & & x & =\tan \theta \\ \therefore & & y & =\sec \theta=\sqrt{1+x^{2}} \end{aligned} $$
On differentiating w.r.t. $x$, we get
$$ \begin{array}{rlrl} \frac{d y}{d x} & =\frac{1}{2 \sqrt{1+x^{2}}} \cdot 2 x \\ \text { At } x=1, & \frac{d y}{d x} & =\frac{1}{\sqrt{2}} \end{array} $$
