Question

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2. 

Answer 1

Mean Value Theorem states that for a function f:[a,b] -> R, if

(a) f is continuous on [a, b]

(b) f is differentiable on (a, b)

ii) f (x) = [x] for x ∈ [– 2, 2]

It is evident that the given function f (x) is not continuous at every integral point.

In particular, f(x) is not continuous at x = −2 and x = 2

⇒ f (x) is not continuous in [−2, 2].

The differentiability of f in (−2, 2) is checked as follows.

Let n be an integer such that n ∈ (−2, 2).