Consider the following statements:

I. The function is continuous at $x=-1$.

II. The function is differentiable at $x=1$.

Which of the statements given above is/are correct?

Calculation:

Given,

The function is defined as:

\( f(x) = \begin{cases} x^3, & \text{if} \, |x| < 1 \\ x^2, & \text{if} \, |x| \geq 1 \end{cases} \)

We are tasked with finding:

\( \lim_{x \to -1} f'(x) \)

Check the left-hand limit for continuity at x = -1 :

\( \lim_{x \to -1^-} f(x) = (-1)^2 = 1 \)

Check the right-hand limit for continuity at x = -1 :

\( \lim_{x \to -1^+} f(x) = (-1)^3 = -1 \)

Since the left-hand limit (L.H.S) and right-hand limit (R.H.S) are not equal, the function is discontinuous at x = -1 .

Check the differentiability at x = 1 :

The left-hand derivative at x = 1  is \(\text{L.H.D} = 3 \) and the right-hand derivative at  x = 1  is \(\text{R.H.D} = 2 \) which means the function is not differentiable at x = 1 

∴ The function is neither continuous at x = -1  nor differentiable at x = 1 . 

Hence, the correct answer is Option 4.