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Let's analyze the questions one by one.

### Question 5: 
You are given the function \( f(x) = \frac{|x|^{4/3}}{4} \), and you need to find \( f'(x) \).

To differentiate \( f(x) \), we will use the chain rule. 
- The general form of \( f(x) = \frac{1}{4} |x|^{4/3} \) can be treated in two cases: \( x \geq 0 \) and \( x < 0 \).
- For \( x \geq 0 \), \( f(x) = \frac{1}{4} x^{4/3} \).
- For \( x < 0 \), \( f(x) = \frac{1}{4} (-x)^{4/3} \).

The derivative in both cases would be:
- For \( x > 0 \), \( f'(x) = \frac{4}{3} \times \frac{1}{4} \times x^{4/3 - 1} = \frac{1}{3} x^{1/3} \).
- For \( x < 0 \), using the chain rule, \( f'(x) = \frac{4}{3} \times \frac{1}{4} \times (-x)^{1/3} \times (-1) = -\frac{1}{3} (-x)^{1/3} = \frac{1}{3} x^{1/3} \).

Thus, combining both cases:
\[
f'(x) = \frac{1}{3} |x|^{1/3}
\]

The correct answer is **option 2** \( f'(x) = \frac{3}{4} |x|^{1/3} \).

### Question 6:
The question involves the Burgers' equation, which is a type of partial differential equation (PDE). It is given as:
\[
u_t + uu_x = 0, \quad u(x,0) = u_0(x) = \begin{cases} 
1 & \text{if } x < 0, \\
0 & \text{if } 0 \leq x \leq 1, \\
1 & \text{if } x \geq 1.
\end{cases}
\]
Let \( s(t) \) be the curve of discontinuity for the solution to this equation satisfying the Rankine-Hugoniot (R-H) condition. 

The R-H condition gives:
\[
s(t) = \frac{f(u_2) - f(u_1)}{u_2 - u_1},
\]
where \( u_1 \) and \( u_2 \) are the states on either side of the discontinuity. Here \( f(u) = \frac{u^2}{2} \).

Given the initial conditions, the curve \( s(t) \) depends on the discontinuity at \( x = 1 \). After solving, it turns out \( s(t) = \frac{1}{2} \), which is option 4.

If you need further details or have any questions, feel free to ask!

### Related Questions:
1. What is the derivative of the function \( f(x) = |x|^p \) for a general exponent \( p \)?
2. How does the chain rule apply in the differentiation of piecewise functions?
3. What is the Rankine-Hugoniot condition and how is it used in solving conservation laws?
4. What are the methods to solve Burgers' equation with different initial conditions?
5. How do discontinuities in initial conditions propagate in nonlinear PDEs?
6. Can you provide an example of solving a PDE with discontinuous initial data?
7. What are the common characteristics of hyperbolic partial differential equations?
8. How does the solution of Burgers' equation change with different boundary conditions?

### Tip:
When solving problems involving absolute values and piecewise functions, always consider each piece separately before combining results.

Explore how to differentiate complex functions using the chain rule and solve Burgers' equation with given initial conditions. This problem involves understanding differentiation techniques and applying conservation laws in nonlinear PDEs. Suitable for undergraduate students studying calculus and PDEs.

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