For the following two (02) items:
Let the function $f(x) = |x - 3| + |x - 4|$ be defined on the interval $[0, 5]$.

Consider the following statements:
I. The function is differentiable at $x = 3$.
II. The function is differentiable at $x = 4$.
Which of the statements given above is/are correct?

The given function is $f(x) = |x - 3| + |x - 4|$ defined on $[0, 5]$.

We can redefine the function by removing the absolute value signs in different intervals:

$f(x) = \begin{cases} -(x - 3) - (x - 4), & 0 \le x < 3 \\ (x - 3) - (x - 4), & 3 \le x < 4 \\ (x - 3) + (x - 4), & 4 \le x \le 5 \end{cases}$

Simplifying this, we get:

$f(x) = \begin{cases} 7 - 2x, & 0 \le x < 3 \\ 1, & 3 \le x < 4 \\ 2x - 7, & 4 \le x \le 5 \end{cases}$

Differentiating $f(x)$ with respect to $x$ in the open intervals:

$f'(x) = \begin{cases} -2, & 0 < x < 3 \\ 0, & 3 < x < 4 \\ 2, & 4 < x < 5 \end{cases}$

At $x = 3$:
Left Hand Derivative (LHD) $= -2$
Right Hand Derivative (RHD) $= 0$
Since LHD $\neq$ RHD, $f(x)$ is not differentiable at $x = 3$.

At $x = 4$:
Left Hand Derivative (LHD) $= 0$
Right Hand Derivative (RHD) $= 2$
Since LHD $\neq$ RHD, $f(x)$ is not differentiable at $x = 4$.

Therefore, neither statement I nor statement II is correct.

Answer: Neither I nor II