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
Mean 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.
c=1.5, where 1.5∈[1,2]