Example 43 - Verify Mean Value Theorem for f(x) = x2 in [2, 4] - Examples

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Example 43 Verify Mean Value Theorem for the function 𝑓(𝑥) = 𝑥2 in the interval [2, 4]. 𝑓(𝑥) = 𝑥2 in interval [2, 4]. Checking conditions for Mean value Theorem Condition 1 𝑓 𝑥﷯= 𝑥﷮2﷯ is continuous at 2 , 4﷯ Since 𝑓(𝑥) is polynomial . it is continuous in 2 , 4﷯ Condition 2 𝑓 𝑥﷯= 𝑥﷮2﷯ is differentiable in 2 , 4﷯ Since 𝑓 𝑥﷯ is a polynomial . it is differentiable is defined 2 , 4﷯ Now, 𝑎 = 2 & 𝑏 = 4 𝑓 𝑎﷯ = 2﷮2﷯ = 4 𝑓 𝑏﷯ = 4﷮2﷯ = 16 𝑓 𝑥﷯ = 𝑥﷮2﷯ 𝑓﷮′﷯ 𝑥﷯= 2𝑥 𝑓﷮′﷯ 𝑐﷯ = 2𝑐 Now 𝑓﷮′﷯ 𝑐﷯ = 𝑓 𝑏﷯ − 𝑓 𝑎﷯﷮𝑏 − 𝑎﷯ 2𝑐 = 16 − 4﷮4 − 2﷯ 2𝑐 = 12﷮2﷯ 2𝑐 = 6 𝑐 = 3 Value of c is 3 which lies between 2 and 4. Hence c = 3 ∈ 𝟐, 𝟒﷯ Hence, Mean value Theorem is satisfied .

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