Last updated at Aug. 13, 2021 by

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Example 43 Verify Mean Value Theorem for the function π(π₯) = π₯2 in the interval [2, 4]. π(π₯) = π₯2 in interval [2, 4]. Checking conditions for Mean value Theorem Condition 1 Since π(π₯) is polynomial . it is continuous β΄ π(π₯) is continuous at (2, 4) Conditions of Mean value theorem π(π₯) is continuous at (π, π) π(π₯) is differentiable at (π , π) If both conditions satisfied, then there exist some c in (π , π) such that πβ²(π) = (π(π) β π(π))/(π β π)Condition 2 Since π(π₯) is a polynomial . it is Differentiable β΄ π(π₯) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c β (2, 4) such that, π^β² (π) = (π(4) β π(2))/(4 β 2) Conditions of Mean value theorem π(π₯) is continuous at (π, π) π(π₯) is differentiable at (π , π) If both conditions satisfied, then there exist some c in (π , π) such that πβ²(π) = (π(π) β π(π))/(π β π) Condition 2 Since π(π₯) is a polynomial . it is Differentiable β΄ π(π₯) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c β (2, 4) such that, π^β² (π) = (π(4) β π(2))/(4 β 2) 2π= (4^2 β 2^2)/2 2π = 12/2 2π = 6 π = π Hence c = 3 β(π, π) Hence, Mean value Theorem is satisfied .

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.