SATHEE: Limit Continuity and Differentiability 4 Question 8

Limit Continuity and Differentiability 4 Question 8

8. For every pair of continuous function $f, g : [0, 1] \to R$ such that $max f (x) : x \in [0, 1] = max g (x) : x \in [0, 1]$ . The correct statement(s) is (are)

(2014 Adv.)

(a) $[f (c)]^{2} + 3 f (c) = [g (c)]^{2} + 3 g (c)$ for some $c \in [0, 1]$

(b) $[f (c)]^{2} + f (c) = [g (c)]^{2} + 3 g (c)$ for some $c \in [0, 1]$

(d) $[f (c)]^{2} = [g (c)]^{2}$ for some $c \in [0, 1]$

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Answer:

Correct Answer: 8. (a, d)

Solution:

The function $f (x) = \tan x$ is not defined at $x = \frac{π}{2}$ , so $f (x)$ is not continuous on $(0, π)$ .

(b) Since, $g (x) = x \sin \frac{1}{x}$ is continuous on $(0, π)$ and the integral function of a continuous function is continuous,

$∴ f (x) = \int_{0}^{x} t \sin \frac{1}{t} d t$ is continuous on $(0, π)$ .

$\begin{aligned} 1, 0 < x \leq \frac{3 π}{4} \\ \frac{2 x}{9}, \frac{3 π}{4} < x < π \end{aligned}$

We have, $lim_{3 π^{-}} f (x) = 1$

$lim_{x \to \frac{3 π^{+}}{4}} f (x) = lim_{x \to \frac{3 π}{4}} 2 \sin \frac{2 x}{9} = 1$

So, $f (x)$ is continuous at $x = 3 π / 4$ .

$\Rightarrow f (x)$ is continuous at all other points.

(d) Finally, $f (x) = \frac{π}{2} \sin (x + π) \Rightarrow f \frac{π}{2} = - \frac{π}{2}$

$\begin{aligned} lim_{x \to {\frac{π}{2}}^{-}} f (x) = lim_{h \to 0} f \frac{π}{2} - h = lim_{h \to 0} \frac{π}{2} \sin \frac{3 π}{2} - h = \frac{π}{2} \\ and lim_{x \to (π / 2)^{+}} f (x) = lim_{h \to 0} f \frac{π}{2} + h \\ = lim_{h \to 0} \frac{π}{2} \sin \frac{3 π}{2} + h = \frac{π}{2} \end{aligned}$

So, $f (x)$ is not continuous at $x = π / 2$ .

Limit Continuity and Differentiability 4 Question 8

8. For every pair of continuous function $f, g : [0, 1] \to R$ such that $max f (x) : x \in [0, 1] = max g (x) : x \in [0, 1]$ . The correct statement(s) is (are)

Answer:

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Limit Continuity and Differentiability 4 Question 8

8. For every pair of continuous function f,g:[0,1]→R such that maxf(x):x∈[0,1]=maxg(x):x∈[0,1]. The correct statement(s) is (are)

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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8. For every pair of continuous function $f, g : [0, 1] \to R$ such that $max f (x) : x \in [0, 1] = max g (x) : x \in [0, 1]$ . The correct statement(s) is (are)