SATHEE: Limit Continuity and Differentiability 7 Question 19

Limit Continuity and Differentiability 7 Question 19

19. The left hand derivative of $f (x) = [x] \sin (π x)$ at $x = k, k$ is an integer, is

(a) $(- 1)^{k} (k - 1) π$

(b) $(- 1)^{k - 1} (k - 1) π$

(d) $(- 1)^{k - 1} k π$

(2001, 2M)

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Answer:

Correct Answer: 19. (b)

Solution:

Given, $f (x) = | \begin{array}{ccc} x^{3} & \sin x & \cos x 6 & - 1 & 0 p & p^{2} & p^{3} \end{array} |$

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

$f^{'} (x) = | \begin{array}{ccc} 3 x^{2} & \cos x & - \sin x \\ 6 & - 1 & 0 \\ p & p^{2} & p^{3} \end{array} | + | \begin{array}{ccc} x^{3} & \sin x & \cos x \\ 0 & 0 & 0 \\ p & p^{2} & p^{3} \end{array} |$

$+ | \begin{array}{ccc} x^{3} & \sin x & \cos x \\ 6 & - 1 & 0 \\ 0 & 0 & 0 \end{array} |$

$\begin{aligned} \Rightarrow & f^{'} (x) & = | \begin{array}{ccc} 3 x^{2} & \cos x & - \sin x \\ 6 & - 1 & 0 \\ p & p^{2} & p^{3} \end{array} | \\ \Rightarrow & f^{''} (x) & = | \begin{array}{ccc} 6 x & - \sin x & - \cos x \\ 6 & - 1 & 0 \\ p & p^{2} & p^{3} \end{array} | + 0 + 0 \end{aligned}$

$\begin{aligned} and f^{'''} (x) = | \begin{array}{ccc} 6 & - \cos x & \sin x \\ 6 & - 1 & 0 \\ p & p^{2} & p^{3} \end{array} | + 0 + 0 \\ ∴ f^{'''} (0) = | \begin{array}{ccc} 6 & - 1 & 0 \\ 6 & - 1 & 0 \\ p & p^{2} & p^{3} \end{array} | = 0 = independent of p \end{aligned}$

Limit Continuity and Differentiability 7 Question 19

19. The left hand derivative of $f (x) = [x] \sin (π x)$ at $x = k, k$ is an integer, is

Answer:

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Limit Continuity and Differentiability 7 Question 19

19. The left hand derivative of f(x)=[x]sin⁡(πx) at x=k,k is an integer, is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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19. The left hand derivative of $f (x) = [x] \sin (π x)$ at $x = k, k$ is an integer, is