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NCERT Class 12- Mathematics: Chapter – 5 Continuity and Differentiability Part 5
Question 17:
Verify Rolle՚s Theorem for the function, in
Answer:
Given . Note that:
(i) The function is continuous in as is a sine function, which is always continuous.
(ii) , exists in , hence is derivable in
(iii) and
Conditions of Rolle՚s Theorem are satisfied. Hence there exists at least one such that . Thus
Question 18:
Verify mean value theorem for the function in .
Answer:
(i) Function is continuous in as product of polynomial functions is a polynomial, which is continuous.
(ii) exists in and hence derivable in .
Thus conditions of mean value theorem are satisfied. Hence, there exists at least one such that
Hence (since other value is not permissible) .
Long Answer (L. A.)
Question 19:
If find the value of so that becomes continuous at .
Answer:
Given,
Therefore,
Thus
If we define , then will become continuous at . Hence for to be continuous at
Question 20:
Show that function given by is discontinuous at
Answer:
The left hand limit of at is given by
Similarly,
Thus therefore, does not exist. Hence is discontinuous at .