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The given function is:

\[
g(x) = \left( 2x - 8 \right)^{2/3} - \frac{1}{x} + x^5
\]

We need to find the **x-values** where \( g(x) \) is continuous but not differentiable.

### Step 1: Analyze Continuity

The function \( g(x) \) consists of three terms:

1. \((2x - 8)^{2/3}\)
2. \(-\frac{1}{x}\)
3. \(x^5\)

We check for continuity for each term:

- \((2x - 8)^{2/3}\): This term is continuous for all \( x \), except when \( 2x - 8 = 0 \) (which gives \( x = 4 \)). At \( x = 4 \), the function remains continuous because fractional exponents (like \( \frac{2}{3} \)) do not cause discontinuity at \( x = 4 \), only non-differentiability.
  
- \(-\frac{1}{x}\): This term is discontinuous at \( x = 0 \) due to division by zero, so the function \( g(x) \) will not be continuous at \( x = 0 \).

- \(x^5\): This term is a simple polynomial and is continuous for all real \( x \).

Thus, **\( g(x) \) is not continuous at \( x = 0 \)** but continuous elsewhere.

### Step 2: Analyze Differentiability

Now, let's check where \( g(x) \) might be continuous but **not differentiable**.

- \((2x - 8)^{2/3}\): This term is not differentiable at \( x = 4 \). Fractional exponents like \( \frac{2}{3} \) cause a "cusp" or a sharp corner, which means the derivative does not exist at \( x = 4 \).

- \(-\frac{1}{x}\) and \(x^5\): These terms are differentiable everywhere except at \( x = 0 \) (where \( g(x) \) is not even continuous).

### Conclusion

The function \( g(x) \) is continuous but **not differentiable at \( x = 4 \)**. There is no continuity or differentiability issue elsewhere except at \( x = 0 \), where the function is not continuous at all.

### Final Answer:
\( g(x) \) is continuous but not differentiable at **\( x = 4 \)**.

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Would you like further details or have any other questions?

Here are 5 related questions you might find helpful:

1. How do fractional exponents affect the differentiability of a function?
2. What are the conditions for a function to be continuous but not differentiable?
3. Can a function be differentiable if it has a sharp corner or cusp?
4. How do we analyze the differentiability of more complex functions with multiple terms?
5. Why does division by zero cause discontinuities in functions?

**Tip:** Always check both continuity and differentiability separately when solving for points of non-differentiability.

Consider the function g(x) = (2x - 8)^(2/3) - 1/x + x^5. Write the x-values for which g(x) is continuous but not differentiable.

Explore how to find the x-values where the function g(x) = (2x - 8)^(2/3) - 1/x + x^5 is continuous but not differentiable. This solution involves key concepts such as continuity, differentiability, and fractional exponents, suitable for high school students in Grades 11-12.

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