Rolles theorem

This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .

1. B.Sc. - ii (mathematics) Advanced calculus Rolle’s Theorem By Preeti Shrivastava

2. Synopsis Rolle’s Theorem Rolle’s theorem’s Example Lagrange’s Mean Value Theorem (first mean value theorem) Example of Lagrange’s mean value theorem Cauchy’s Mean Value Theorem(second mean value theorem) Example of Cauchy’s mean value theorem

3. Rolle’s Theorem

4. Statement:- If f(x) is a function of the variable x such that :- f(x) is continuous in the closed interval [a,b]. f(x) is differentiable for every point in the open interval (a,b). f(a) = f(b), then there is at least one point c such that f’(c) =0.

5. Proof:- We need to know Close Interval and Open Interval: Open Interval:- An open interval is an interval that does not include its two mid- point .The open interval {x:a<x<b} is denoted by (a,b). Close Interval:- A closed interval is an interval that include all its limit point. A finite number a and b then the interval {x:a<x<b} is denoted by [a,b]. Case 1:- If f(x) is constant then, f(x) = k (constant) f(a) = k f(b) = k

6. so, f(a) = f(b) f(a) f(b) since, f(x) =k a b therefore f’(x) =0 and f’(c) =0 Case 2:- If function f(x) will first increase and then decreases. Then at some point C consequently f(x) will be maximum. Then f(c-h) < f (c) and f( c+h ) < f(c) f(c-h) – f(c) < 0 and f( c+h ) - f(c) < 0 Taking limit on both sides: and

7. Lagrange’s Mean Value Theorem

8. Squaring on both side Hence Lagrange’s Theorem is Verified.

9. Cauchy’ mean value theorem

10. Statement:- If two function f(x) and g(x) are:- Continuous in the Closed Interval [a,b]. Differentiable in the Open Interval [a,b]. Then there is at least one point c such that:- where a<c<b Proof:- Now consider a function F(x) defined by:- F(x)= f(x)+ A(x) ……….(I) Where A is a constant to be determined

11. such that : F(a) = F(b) ……….(II) Since f(x) are continuous in closed interval [a,b]. So, F(x) is also continuous in closed interval [a,b]. Again, f(x) are differentiable in Open interval (a,b). So, F(x) is also differentiable in the open interval (a,b) Then all condition of Rolle’s Theorem will be satisfied .

12. Hence by the Rolle’s Theorem there at least one Point C in open interval (a,b) such that F’(c) =0 ……(III) from Equation (1):-  F(x) = f(x) + A ∅(x) F(a) = f(a)+ A ∅(a) ……..(IV) F(b) = f(b)+ A ∅(b) ………(V) Now from Equation (II), we get:- F(a) =F(b)

13. Example:- Verify Cauchy’s mean value theorem for function and in interval [1,2]

14. THANK YOU