Question

# Examine the applicability of Mean Value Theorem for the following function.f(x)=x2−1 for xϵ[1,2]

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Solution

## Mean Value Theorem holds for a function f:[a,b]→R, if following two conditions holds(i) f is continuous on [a,b](ii) f is differentiable on (a,b)Then, there exists some c∈(a,b) such that f′(c)=f(b)−f(a)b−aMean Value Theorem is not applicable to those functions that do not satisfy any of the two conditions of the hypothesis.f(x)=x2−1 for x∈ [1,2] Since, f(x) is a polynomial function. Polynomial functions are continuous and differentiable everywhere.So, f(x) is continuous in [1,2] and is differentiable in (1,2). Hence, f satisfies the conditions of the hypothesis of Mean Value Theorem. Hence, Mean Value Theorem is applicable for given function f(x)It can be proved as follows.f(1)=12−1=0, f(2)=22−1=3∴f(b)−f(a)b−a=f(2)−f(1)2−1=3−01=3Also, f′(x)=2x∴f′(c)=3⇒2c=3⇒c=32c=1.5, where 1.5∈[1,2]

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