Let y = f(x), choose the correct statement:
By BYJU'S Exam Prep
Updated on: October 17th, 2023
(1) f is differentiable in [a, b] if f'(x) exist for all x ϵ (a, b)
(2) f is differentiable in (a, b) if it is differentiable at x = a and x = b
(3) f is differentiable in [a, b] if f'(a + 0) and f'(a – 0) exist
(4) f is differentiable in (a, b) if it is differentiable in x ϵ (a, b)
The statement f is differentiable in (a, b) if it is differentiable in x ϵ (a, b) is correct. Differentiability of an interval
- In (a, b), y = f(x) is differentiable at x ϵ (a, b)
- In [a, b], y = f(x) is differentiable if f’(x) exist and f’ (a + 0), f’ (a – 0) exist for all x ϵ (a, b)
Let us calculate:
Option 1 is not true because the function is differentiable in [a, b] if f'(a + 0), f'(a – 0) and f'(x) exist for all x ϵ (a, b).
Option 2 is not true because the function is differentiable in (a, b), it has to be differentiable not only at x = a and x = b but also in each point of (a, b).
Option 3 is not true because the function is differentiable in [a, b] if f'(x) exist for x ϵ (a, b) and f'(a + 0), f'(a – 0) exist.
Option 4 is true as the function is differentiable in (a, b) if it is differentiable in x ϵ (a, b)
Therefore, the correct statement is that f is differentiable in (a, b) if it is differentiable in x ϵ (a, b).
Summary:
Let y = f(x), choose the correct statement: (1) f is differentiable in [a, b] if f'(x) exist for all x ϵ (a, b) (2) f is differentiable in (a, b) if it is differentiable at x = a and x = b (3) f is differentiable in [a, b] if f'(a + 0) and f'(a – 0) exist (4) f is differentiable in (a, b) if it is differentiable in x ϵ (a, b)
Let y = f(x), the correct statement is f is differentiable in (a, b) if it is differentiable in x ϵ (a, b).
