Check sibling questions

Example 43 - Examples

Example 43 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Example 43 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

This video is only available for Teachoo black users

Introducing your new favourite teacher - Teachoo Black, at only β‚Ή83 per month


Transcript

Example 43 Verify Mean Value Theorem for the function 𝑓(π‘₯) = π‘₯2 in the interval [2, 4]. 𝑓(π‘₯) = π‘₯2 in interval [2, 4]. Checking conditions for Mean value Theorem Condition 1 Since 𝑓(π‘₯) is polynomial . it is continuous ∴ 𝑓(π‘₯) is continuous at (2, 4) Conditions of Mean value theorem 𝑓(π‘₯) is continuous at (π‘Ž, 𝑏) 𝑓(π‘₯) is differentiable at (π‘Ž , 𝑏) If both conditions satisfied, then there exist some c in (π‘Ž , 𝑏) such that 𝑓′(𝑐) = (𝑓(𝑏) βˆ’ 𝑓(π‘Ž))/(𝑏 βˆ’ π‘Ž)Condition 2 Since 𝑓(π‘₯) is a polynomial . it is Differentiable ∴ 𝑓(π‘₯) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c ∈ (2, 4) such that, 𝑓^β€² (𝑐) = (𝑓(4) βˆ’ 𝑓(2))/(4 βˆ’ 2) Conditions of Mean value theorem 𝑓(π‘₯) is continuous at (π‘Ž, 𝑏) 𝑓(π‘₯) is differentiable at (π‘Ž , 𝑏) If both conditions satisfied, then there exist some c in (π‘Ž , 𝑏) such that 𝑓′(𝑐) = (𝑓(𝑏) βˆ’ 𝑓(π‘Ž))/(𝑏 βˆ’ π‘Ž) Condition 2 Since 𝑓(π‘₯) is a polynomial . it is Differentiable ∴ 𝑓(π‘₯) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c ∈ (2, 4) such that, 𝑓^β€² (𝑐) = (𝑓(4) βˆ’ 𝑓(2))/(4 βˆ’ 2) 2𝑐= (4^2 βˆ’ 2^2)/2 2𝑐 = 12/2 2𝑐 = 6 𝒄 = πŸ‘ Hence c = 3 ∈(𝟐, πŸ’) Hence, Mean value Theorem is satisfied .

Davneet Singh's photo - Teacher, Engineer, Marketer

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.