Question

Examine that Rolle’s Theorem is applicable to the following function in the given intervals, justify your

1. f(x) = [x], x ∈ [5, 9]
2. f(x) = x² – 1, x ∈ [1, 2]

Answer 1

 1. The function f(x) = [x] is not differentiable and continuous at integral values. So in the given interval [5, 9] the function is neither differentiable nor continuous at x = 6, 7 ,8. Therefore Rolle’s Theorem is not applicable.

2. The function f(x) = x² – 1 is a polynomial function so differentiable and continuous.
f(1) = 1 – 1 = 0, f(2) = 4 – 1 = 3
f(1) ≠ f(2) . Therefore Rolle’s Theorem is not applicable.