NCERT Class 12-Mathematics: Chapter – 5 Continuity and Differentiability Part 5 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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 .