Examples

Chapter 5 Class 12 Continuity and Differentiability
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Example 43 Verify Mean Value Theorem for the function ๐(๐ฅ) = ๐ฅ2 in the interval [2, 4]. ๐(๐ฅ) = ๐ฅ2 in interval [2, 4]. Checking conditions for Mean value Theorem Condition 1 Since ๐(๐ฅ) is polynomial . it is continuous โด ๐(๐ฅ) is continuous at (2, 4) Conditions of Mean value theorem ๐(๐ฅ) is continuous at (๐, ๐) ๐(๐ฅ) is differentiable at (๐ , ๐) If both conditions satisfied, then there exist some c in (๐ , ๐) such that ๐โฒ(๐) = (๐(๐) โ ๐(๐))/(๐ โ ๐)Condition 2 Since ๐(๐ฅ) is a polynomial . it is Differentiable โด ๐(๐ฅ) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c โ (2, 4) such that, ๐^โฒ (๐) = (๐(4) โ ๐(2))/(4 โ 2) Conditions of Mean value theorem ๐(๐ฅ) is continuous at (๐, ๐) ๐(๐ฅ) is differentiable at (๐ , ๐) If both conditions satisfied, then there exist some c in (๐ , ๐) such that ๐โฒ(๐) = (๐(๐) โ ๐(๐))/(๐ โ ๐) Condition 2 Since ๐(๐ฅ) is a polynomial . it is Differentiable โด ๐(๐ฅ) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c โ (2, 4) such that, ๐^โฒ (๐) = (๐(4) โ ๐(2))/(4 โ 2) 2๐= (4^2 โ 2^2)/2 2๐ = 12/2 2๐ = 6 ๐ = ๐ Hence c = 3 โ(๐, ๐) Hence, Mean value Theorem is satisfied .