SATHEE: Limit Continuity and Differentiability 7 Question 32

Limit Continuity and Differentiability 7 Question 32

33. Let $f : [a, b] \to [1, \infty)$ be a continuous function and

$g : R \to R be defined as g (x) = \begin{array}{ccc} 0, & if & x < a \\ \int_{a}^{x} f (t) d t, & if & a \leq x \leq b . \\ \int_{a}^{b} f (t) d t, & if & x > b \end{array}$

Then,

(2013)

(a) $g (x)$ is continuous but not differentiable at $a$

(b) $g (x)$ is differentiable on $R$

(d) $g (x)$ is continuous and differentiable at either $a$ or $b$ but not both

$- x - \frac{π}{2}, x \leq - \frac{π}{2}$

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Answer:

Correct Answer: 33. $x = 0$

Solution:

Given, $(a + b x) e^{y / x} = x \Rightarrow y = x \log \frac{x}{a + b x}$

$\Rightarrow y = x [\log (x) - \log (a + b x)]$

On differentiating both sides, we get

$\begin{aligned} \frac{d y}{d x} & = x \frac{1}{x} - \frac{b}{a + b x} + 1 [\log (x) - \log (a + b x)] \\ \Rightarrow x \frac{d y}{d x} & = x^{2} \frac{a}{x (a + b x)} + y \\ \Rightarrow x y_{1} & = \frac{a x}{a + b x} + y \end{aligned}$

Again, differentiating both sides, we get

$\begin{aligned} \Rightarrow & x y_{2} + y_{1} & = a \frac{(a + b x) \cdot 1 - x \cdot b}{(a + b x)^{2}} + y_{1} \\ \Rightarrow & x^{3} y_{2} & = \frac{a^{2} x^{2}}{(a + b x)^{2}} \\ \Rightarrow & x^{3} y_{2} & = \frac{a x}{(a + b x)} \\ \Rightarrow & x^{3} y_{2} & = {(x y_{1} - y)}^{2} \\ \Rightarrow & x^{3} \frac{d^{2} y}{d x^{2}} & = x \frac{d y}{d x} - y \end{aligned}$

[from Eq. (ii)]

Limit Continuity and Differentiability 7 Question 32

33. Let $f : [a, b] \to [1, \infty)$ be a continuous function and

Answer:

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Limit Continuity and Differentiability 7 Question 32

33. Let f:[a,b]→[1,∞) be a continuous function and

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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33. Let $f : [a, b] \to [1, \infty)$ be a continuous function and