SATHEE: Limit Continuity and Differentiability 7 Question 3

Limit Continuity and Differentiability 7 Question 3

3. Let $f (x) = 15 - | x - 10 |; x \in R$ . Then, the set of all values of $x$ , at which the function, $g (x) = f (f (x))$ is not differentiable, is

(2019 Main, 9 April I)

(a) $5, 10, 15, 20$

(b) $5, 10, 15$

(d) $10, 15$

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Answer:

Correct Answer: 3. (a)

Solution:

Key Idea Differentiating the given equation twice w.r.t. ’ $x$ ‘.

Given equation is

$e^{y} + x y = e$

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

$\begin{aligned} e^{y} \frac{d y}{d x} + x \frac{d y}{d x} + y & = 0 \\ \frac{d y}{d x} & = - \frac{y}{e^{y} + x} \end{aligned}$

Again differentiating Eq. (ii) w.r.t. ’ $x$ ‘, we get

$e^{y} \frac{d^{2} y}{d x^{2}} + e^{y} \frac{d y}{d x} + x \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} + \frac{d y}{d x} = 0$

Now, on putting $x = 0$ in Eq. (i), we get

$\begin{aligned} e^{y} & = e^{1} \\ y & = 1 \end{aligned}$

On putting $x = 0, y = 1$ in Eq. (iii), we get

$\frac{d y}{d x} = - \frac{1}{e + 0} = - \frac{1}{e}$

Now, on putting $x = 0, y = 1$ and $\frac{d y}{d x} = - \frac{1}{e}$ in

Eq. (iv), we get

$\begin{aligned} e^{1} \frac{d^{2} y}{d x^{2}} + e^{1} - \frac{1}{e} + 0 \frac{d^{2} y}{d x^{2}} + - \frac{1}{e} + - \frac{1}{e} = 0 \\ {\Rightarrow \frac{d^{2} y}{d x^{2}} |}_{(0, 1)} = \frac{1}{e^{2}} \\ So, \frac{d y}{d x}, \frac{d^{2} y}{d x^{2}} at (0, 1) is - \frac{1}{e}, \frac{1}{e^{2}} \end{aligned}$

Limit Continuity and Differentiability 7 Question 3

3. Let $f (x) = 15 - | x - 10 |; x \in R$ . Then, the set of all values of $x$ , at which the function, $g (x) = f (f (x))$ is not differentiable, is

Answer:

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Limit Continuity and Differentiability 7 Question 3

3. Let f(x)=15−|x−10|;x∈R. Then, the set of all values of x, at which the function, g(x)=f(f(x)) is not differentiable, is

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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3. Let $f (x) = 15 - | x - 10 |; x \in R$ . Then, the set of all values of $x$ , at which the function, $g (x) = f (f (x))$ is not differentiable, is