https://file.aiquickdraw.com/mathbot/file/172304474632237opxvvz.jpg

It seems that the image is the same as the one previously uploaded. The analysis and solution I provided earlier still hold. To summarize:

### Question 5:
- You need to differentiate the function \( f(x) = \frac{|x|^{4/3}}{4} \).
- The correct derivative is \( f'(x) = \frac{1}{3} |x|^{1/3} \), which corresponds to **option 2** \( \frac{3}{4} |x|^{1/3} \) based on the given options.

### Question 6:
- The Burgers' equation is given with initial conditions that result in a discontinuity.
- The curve \( s(t) \), representing the location of the discontinuity, is calculated using the Rankine-Hugoniot condition.
- The correct answer for \( s(t) \) is **\( \frac{1}{2} \)**, corresponding to option 4.

Let me know if you need further clarification or details on these solutions! 

### Related Questions:
1. How do you compute the derivative of absolute value functions?
2. What are some common techniques for differentiating piecewise functions?
3. What are shock waves and how are they related to the Burgers' equation?
4. What role does the Rankine-Hugoniot condition play in solving hyperbolic PDEs?
5. How do different initial conditions affect the solution to Burgers' equation?
6. What is the significance of the characteristic lines in solving PDEs?
7. Can you explain how to solve the Burgers' equation using the method of characteristics?
8. What are some real-world applications of the Burgers' equation?

### Tip:
When working with PDEs, always check the nature of the equation (elliptic, parabolic, or hyperbolic) to determine the appropriate solution method.

Explore detailed solutions for differentiating the function f(x) = |x|^(4/3)/4 and solving Burgers' equation with initial conditions causing a discontinuity. Includes explanations of key concepts like derivatives of absolute value functions and the Rankine-Hugoniot condition.

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