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Chapter 5 Class 12 Continuity and Differentiability | 10 questions

Timed test. Answer every question before time runs out. Correct answers and solutions appear after submission.
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Question 1 of 10
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Question 1 of 10
Rolle s Theorem
Rolle s Theorem
Rolle's Theorem: Suppose following three condition hold for function \(\mathrm{y}=f(\mathrm{x})\) :
-1. function is defined and continuous on closed interval \([\mathrm{a}, \mathrm{b}]\);
-2. exists finite derivative \(f^{\prime}(x)\) on interval \((a, b)\);
-3. \(f(\mathrm{a})=f(\mathrm{~b})\).
then there exists point \(\mathrm{c}(\mathrm{a}<\mathrm{c}<\mathrm{b})\) such that \(f^{\prime}(\mathrm{c})=0\).

Based on the above information, answer any four of the following questions.
(i) Rolle's theorem is not applicable for the function \(\mathrm{f}(\mathrm{x})=\tan \mathrm{x}\) in \([0, \pi]\) because \(\_\_\_\_\) .
Select one option. Answers are shown after the test.
Question 2 of 10
Rolle s Theorem
Rolle s Theorem
Rolle's Theorem: Suppose following three condition hold for function \(\mathrm{y}=f(\mathrm{x})\) :
-1. function is defined and continuous on closed interval \([\mathrm{a}, \mathrm{b}]\);
-2. exists finite derivative \(f^{\prime}(x)\) on interval \((a, b)\);
-3. \(f(\mathrm{a})=f(\mathrm{~b})\).
then there exists point \(\mathrm{c}(\mathrm{a}<\mathrm{c}<\mathrm{b})\) such that \(f^{\prime}(\mathrm{c})=0\).

Based on the above information, answer any four of the following questions.
The value of c satisfying Rolle's theorem for the function \(\mathrm{g}(\mathrm{x})=\sin \mathrm{x}\) in \([0, \pi]\) is \(\_\_\_\_\) .
Select one option. Answers are shown after the test.
Question 3 of 10
Rolle s Theorem
Rolle s Theorem
Rolle's Theorem: Suppose following three condition hold for function \(\mathrm{y}=f(\mathrm{x})\) :
-1. function is defined and continuous on closed interval \([\mathrm{a}, \mathrm{b}]\);
-2. exists finite derivative \(f^{\prime}(x)\) on interval \((a, b)\);
-3. \(f(\mathrm{a})=f(\mathrm{~b})\).
then there exists point \(\mathrm{c}(\mathrm{a}<\mathrm{c}<\mathrm{b})\) such that \(f^{\prime}(\mathrm{c})=0\).

Based on the above information, answer any four of the following questions.
The value of c satisfying Rolle's theorem for the function \(\mathrm{h}(\mathrm{x})= \cos x\) in \([0,2 \pi]\) is \(\_\_\_\_\) .
Select one option. Answers are shown after the test.
Question 4 of 10
Rolle s Theorem
Rolle s Theorem
Rolle's Theorem: Suppose following three condition hold for function \(\mathrm{y}=f(\mathrm{x})\) :
-1. function is defined and continuous on closed interval \([\mathrm{a}, \mathrm{b}]\);
-2. exists finite derivative \(f^{\prime}(x)\) on interval \((a, b)\);
-3. \(f(\mathrm{a})=f(\mathrm{~b})\).
then there exists point \(\mathrm{c}(\mathrm{a}<\mathrm{c}<\mathrm{b})\) such that \(f^{\prime}(\mathrm{c})=0\).

Based on the above information, answer any four of the following questions.
The value of c satisfying Rolle's theorem for the function \(p(x)=\sin x+\cos x\) in \([0, \pi]\) is \(\_\_\_\_\) .
Select one option. Answers are shown after the test.
Question 5 of 10
Rolle s Theorem
Rolle s Theorem
Rolle's Theorem: Suppose following three condition hold for function \(\mathrm{y}=f(\mathrm{x})\) :
-1. function is defined and continuous on closed interval \([\mathrm{a}, \mathrm{b}]\);
-2. exists finite derivative \(f^{\prime}(x)\) on interval \((a, b)\);
-3. \(f(\mathrm{a})=f(\mathrm{~b})\).
then there exists point \(\mathrm{c}(\mathrm{a}<\mathrm{c}<\mathrm{b})\) such that \(f^{\prime}(\mathrm{c})=0\).

Based on the above information, answer any four of the following questions.
Rolle's theorem is not applicable for the function \(f(x)=|x|\) in [-2, 2] because \(\_\_\_\_\) .
Select one option. Answers are shown after the test.
Question 6 of 10
Ms Remka of
Ms Remka of
Ms. Remka of city school is teaching chain rule to her students with the help of a flow-chart The chain rule says that if \(h\) and \(g\) are functions and \(f(x)=g(h(x))\), then
Based on the above information, answer any four of the following questions.
Let \(f(x)=\sin x\) and \(g(x)=x^3\)
\(f \circ g(\mathrm{x})=\) \(\_\_\_\_\) .
Select one option. Answers are shown after the test.
Question 7 of 10
Ms Remka of
Ms Remka of
Ms. Remka of city school is teaching chain rule to her students with the help of a flow-chart The chain rule says that if \(h\) and \(g\) are functions and \(f(x)=g(h(x))\), then
Based on the above information, answer any four of the following questions.
Let \(f(x)=\sin x\) and \(g(x)=x^3\)
\(g \circ f(x)=\) \(\_\_\_\_\) .
Select one option. Answers are shown after the test.
Question 8 of 10
Ms Remka of
Ms Remka of
Ms. Remka of city school is teaching chain rule to her students with the help of a flow-chart The chain rule says that if \(h\) and \(g\) are functions and \(f(x)=g(h(x))\), then
Based on the above information, answer any four of the following questions.
Let \(f(x)=\sin x\) and \(g(x)=x^3\)
$$ \frac{d}{d x}\left(\sin ^3 x\right)= $$

\(\_\_\_\_\) .
Select one option. Answers are shown after the test.
Question 9 of 10
Ms Remka of
Ms Remka of
Ms. Remka of city school is teaching chain rule to her students with the help of a flow-chart The chain rule says that if \(h\) and \(g\) are functions and \(f(x)=g(h(x))\), then
Based on the above information, answer any four of the following questions.
Let \(f(x)=\sin x\) and \(g(x)=x^3\)
$$ \frac{d}{d x}\left(\sin x^3\right) $$

\(\_\_\_\_\) .
Select one option. Answers are shown after the test.
Question 10 of 10
Ms Remka of
Ms Remka of
Ms. Remka of city school is teaching chain rule to her students with the help of a flow-chart The chain rule says that if \(h\) and \(g\) are functions and \(f(x)=g(h(x))\), then
Based on the above information, answer any four of the following questions.
Let \(f(x)=\sin x\) and \(g(x)=x^3\)
\(\frac{d}{d x}(\sin 2 \mathrm{x})\) at \(\mathrm{x}=\frac{\pi}{2}\) is \(\_\_\_\_\) .
Select one option. Answers are shown after the test.