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Ex 5.8, 4 - Verify Mean Value Theorem f(x) = x2 - 4x - 3

Ex 5.8, 4 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.8, 4 - Chapter 5 Class 12 Continuity and Differentiability - Part 3
Ex 5.8, 4 - Chapter 5 Class 12 Continuity and Differentiability - Part 4
Ex 5.8, 4 - Chapter 5 Class 12 Continuity and Differentiability - Part 5

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Ex 5.8, 4 Verify Mean Value Theorem, if 𝑓 (π‘₯) = π‘₯2 – 4π‘₯ – 3 in the interval [π‘Ž, 𝑏], where π‘Ž= 1 π‘Žπ‘›π‘‘ 𝑏= 4 𝑓 (π‘₯) = π‘₯2 – 4π‘₯ – 3 π‘₯∈[π‘Ž, 𝑏] where a = 1 & b = 4 Mean Value Theorem satisfied if Condition 1 𝑓(π‘₯) is continuous 𝑓(π‘₯)=π‘₯2 – 4π‘₯ – 3 𝑓(π‘₯) is a polynomial & Every polynomial function is continuous β‡’ 𝑓(π‘₯) is continuous at π‘₯∈[1, 4] Conditions of Mean value theorem 𝑓(π‘₯) is continuous at (π‘Ž , 𝑏) 𝑓(π‘₯) is derivable 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) Conditions of Mean value theorem 𝑓(π‘₯) is continuous at (π‘Ž , 𝑏) 𝑓(π‘₯) is derivable 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 If 𝑓(π‘₯) is differentiable 𝑓(π‘₯) = π‘₯2 – 4π‘₯ – 3 𝑓(π‘₯) is a polynomial & Every polynomial function is differentiable β‡’ 𝑓(π‘₯) is differentiable at π‘₯∈[1, 4] Condition 3 𝑓(π‘₯) = π‘₯2 – 4π‘₯ – 3 𝑓^β€² (π‘₯) = 2π‘₯βˆ’4 𝑓^β€² (𝑐) = 2π‘βˆ’4 Conditions of Mean value theorem 𝑓(π‘₯) is continuous at (π‘Ž , 𝑏) 𝑓(π‘₯) is derivable 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) 𝑓(π‘Ž) = 𝑓(1) = (1)^2βˆ’4(1)βˆ’3 = 1 βˆ’ 4 βˆ’ 3 = βˆ’6 𝑓(𝑏) = 𝑓(4) = (4)^2βˆ’4(4)βˆ’3 = 16 βˆ’ 16 βˆ’ 3 = βˆ’ 3 By Mean Value Theorem 𝑓^β€² (𝑐) = (𝑓(𝑏) βˆ’ 𝑓(π‘Ž))/(𝑏 βˆ’ π‘Ž) 𝑓^β€² (𝑐) = (βˆ’3 βˆ’ (βˆ’6))/(4 βˆ’ 1) 𝑓^β€² (𝑐) = (βˆ’3 + 6)/3 𝑓^β€² (𝑐) = 3/3 𝑓^β€² (𝑐) = 1 2c βˆ’ 4 = 1 2c = 1 + 4 2c = 5 c = 5/2 Value of c = 5/2 which is lies between (1, 4) c = πŸ“/𝟐∈(𝟏, πŸ’) Hence Mean Value Theorem satisfied

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.