Let y = f(x), choose the correct statement:

By Ritesh|Updated : October 25th, 2022

(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

  1. In (a, b), y = f(x) is differentiable at x ϵ (a, b)
  2. 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).

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