SATHEE: Limit Continuity and Differentiability 7 Question 34

Limit Continuity and Differentiability 7 Question 34

35. Let $f : R \to R$ be a function such that $f (x + y) = f (x) + f (y), \forall x, y \in R$ . If $f (x)$ is differentiable at $x = 0$ , then

(2011)

(a) $f (x)$ is differentiable only in a finite interval containing zero

(b) $f (x)$ is continuous for all $x \in R$

(d) $f (x)$ is differentiable except at finitely many points

Show Answer

Answer:

Correct Answer: 35. $(a = 1)$

Solution:

Since, $y = e^{x \sin x^{3}} + (\tan x)^{x}$ , then

$y = u + v$ , where $u = e^{x \sin x^{3}}$ and $v = (\tan x)^{x}$

$\Rightarrow \frac{d y}{d x} = \frac{d u}{d x} + \frac{d v}{d x}$

Here, $u = e^{x \sin x^{3}}$ and $\log v = x \log (\tan x)$

On differentiating both sides w.r.t. $x$ , we get

$\begin{aligned} \frac{d u}{d x} & = e^{x \sin x^{3}} \cdot (3 x^{3} \cos x^{3} + \sin x^{3}) \\ and \frac{1}{v} \cdot \frac{d v}{d x} & = \frac{x \cdot \sec^{2} x}{\tan x} + \log (\tan x) \\ \frac{d v}{d x} & = (\tan x)^{x} [2 x \cdot cosec (2 x) + \log (\tan x)] \dots (iii) \end{aligned}$

From Eqs. (i), (ii) and (iii), wet get

$\frac{d y}{d x} = e^{x \sin x^{3}} (3 x^{3} \cdot \cos x^{3} + \sin x^{3}) + (\tan x)^{x}$

$[2 x cosec 2 x + \log (\tan x)]$

Limit Continuity and Differentiability 7 Question 34

35. Let $f : R \to R$ be a function such that $f (x + y) = f (x) + f (y), \forall x, y \in R$ . If $f (x)$ is differentiable at $x = 0$ , then

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Limit Continuity and Differentiability 7 Question 34

35. Let f:R→R be a function such that f(x+y)=f(x)+f(y),∀x,y∈R. If f(x) is differentiable at x=0, then

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

35. Let $f : R \to R$ be a function such that $f (x + y) = f (x) + f (y), \forall x, y \in R$ . If $f (x)$ is differentiable at $x = 0$ , then