SATHEE: Limit Continuity and Differentiability 7 Question 31

Limit Continuity and Differentiability 7 Question 31

32. Let $f, g : [- 1, 2] \to R$ be continuous functions which are twice differentiable on the interval $(- 1, 2)$ . Let the values of $f$ and $g$ at the points $- 1, 0$ and 2 be as given in the following table:

	$x = - 1$	$x = 0$	$x = 2$
$f (x)$	3	6	0
$g (x)$	0	1	-1

In each of the intervals $(- 1, 0)$ and $(0, 2)$ , the function $(f - 3 g)^{''}$ never vanishes. Then, the correct statement(s) is/are

(2015 Adv.) (a) $f^{'} (x) - 3 g^{'} (x) = 0$ has exactly three solutions in $(- 1, 0) \cup (0, 2)$

(b) $f^{'} (x) - 3 g^{'} (x) = 0$ has exactly one solution in $(- 1, 0)$

(d) $f^{'} (x) - 3 g^{'} (x) = 0$ has exactly two solutions in $(- 1, 0)$ and exactly two solutions in $(0, 2)$

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Answer:

Correct Answer: 32. $f^{'} (0^{+}) = 0, f^{'} (0^{-}) = 1$

Solution:

Given, $y = (\log_{\cos x} \sin x) \cdot {(\log_{\sin x} \cos x)}^{- 1} + \sin^{- 1} \frac{2 x}{1 + x^{2}}$

$\begin{aligned} ∴ y = {\frac{\log_{e} (\sin x)}{\log_{e} (\cos x)}}^{2} + \sin^{- 1} \frac{2 x}{1 + x^{2}} \\ (\log_{e} (\cos x) \cdot \cot x \\ \Rightarrow \frac{d y}{d x} = 2 \frac{\log_{e} (\sin x)}{\log_{e} (\cos x)} \cdot \frac{+ \log_{e} (in x) \cdot \tan x)}{{\log_{e} (\cos x)}^{2}} + \frac{2}{1 + x^{2}} \end{aligned}$

$\begin{aligned} \Rightarrow \frac{d y}{d x} \int_{x = \frac{π}{4}} & = 21 \cdot \frac{2 \cdot \log \frac{1}{\sqrt{2}}}{\log \frac{1}{\sqrt{2}}} + \frac{2}{1 + \frac{π^{2}}{16}} \\ = - \frac{8}{\log_{e} 2} + \frac{32}{16 + π^{2}} \end{aligned}$

Limit Continuity and Differentiability 7 Question 31

32. Let $f, g : [- 1, 2] \to R$ be continuous functions which are twice differentiable on the interval $(- 1, 2)$ . Let the values of $f$ and $g$ at the points $- 1, 0$ and 2 be as given in the following table:

Answer:

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Limit Continuity and Differentiability 7 Question 31

32. Let f,g:[−1,2]→R be continuous functions which are twice differentiable on the interval (−1,2). Let the values of f and g at the points −1,0 and 2 be as given in the following table:

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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32. Let $f, g : [- 1, 2] \to R$ be continuous functions which are twice differentiable on the interval $(- 1, 2)$ . Let the values of $f$ and $g$ at the points $- 1, 0$ and 2 be as given in the following table: