SATHEE: Limit Continuity and Differentiability 4 Question 9

Limit Continuity and Differentiability 4 Question 9

9. For every integer $n$ , let $a_{n}$ and $b_{n}$ be real numbers. Let function $f : R \to R$ be given by

$f (x) = \begin{array}{ll} a_{n} + \sin π x, & for x \in [2 n, 2 n + 1] b_{n} + \cos π x, & for x \in (2 n - 1, 2 n) \end{array}$

for all integers $n$ .

If $f$ is continuous, then which of the following hold(s) for all $n$ ?

(2012)

(a) $a_{n - 1} - b_{n - 1} = 0$

(b) $a_{n} - b_{n} = 1$

(d) $a_{n - 1} - b_{n} = - 1$

Fill in the Blank

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Answer:

Correct Answer: 9. $f (x) = \sqrt{4 - x^{2}}$

Solution:

We have, for $- 1 < x < 1$

$\Rightarrow 0 \leq x \sin π x \leq 1 / 2$

$∴ [x \sin π x] = 0$

Also, $x \sin π x$ becomes negative and numerically less than 1 when $x$ is slightly greater than 1 and so by definition of $[x]$ .

$f (x) = [x \sin π x] = - 1, when 1 < x < 1 + h$

Thus, $f (x)$ is constant and equal to 0 in the closed interval $[- 1, 1]$ and so $f (x)$ is continuous and differentiable in the open interval $(- 1, 1)$ .

At $x = 1, f (x)$ is discontinuous, since $lim_{h \to 0} (1 - h) = 0$

and $lim_{h \to 0} (1 + h) = - 1$

$∴ f (x)$ is not differentiable at $x = 1$ .

Hence, (a), (b) and (d) are correct answers.

Limit Continuity and Differentiability 4 Question 9

9. For every integer $n$ , let $a_{n}$ and $b_{n}$ be real numbers. Let function $f : R \to R$ be given by

Fill in the Blank

Answer:

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Limit Continuity and Differentiability 4 Question 9

9. For every integer n, let an and bn be real numbers. Let function f:R→R be given by

Fill in the Blank

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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9. For every integer $n$ , let $a_{n}$ and $b_{n}$ be real numbers. Let function $f : R \to R$ be given by