. [0,3] R defined by f(x) = (x - 2) ^ (1/5) lmvt is applicable 

Let's analyze the given function \( f(x) = (x - 2)^{1/5} \) over the interval \([0, 3]\) to determine if the **Lagrange Mean Value Theorem (LMVT)** is applicable.

The **Mean Value Theorem (MVT)** requires two conditions to be satisfied for a function \( f \) on an interval \( [a, b] \):

1. **Continuity**: \( f(x) \) must be continuous on the closed interval \([a, b]\).
2. **Differentiability**: \( f(x) \) must be differentiable on the open interval \( (a, b) \).

### Step 1: Continuity on \([0, 3]\)
The function \( f(x) = (x - 2)^{1/5} \) is a real-valued function. To check its continuity, note that the function is continuous everywhere except at the point where \( x - 2 = 0 \), i.e., \( x = 2 \). The function is continuous over the entire real line since fractional exponents like \( (x-2)^{1/5} \) are defined for all real numbers. Therefore, \( f(x) \) is continuous on the entire interval \([0, 3]\).

### Step 2: Differentiability on \( (0, 3) \)
Now, we need to check differentiability on the open interval \( (0, 3) \).

- The derivative of \( f(x) \) is:
  \[
  f'(x) = \frac{1}{5} (x - 2)^{-4/5}
  \]
  This derivative is not defined at \( x = 2 \), because the exponent \( -4/5 \) causes a division by zero when \( x = 2 \). Therefore, \( f(x) \) is not differentiable at \( x = 2 \), even though it is differentiable everywhere else in the interval.

### Conclusion:
Since the function \( f(x) = (x - 2)^{1/5} \) is not differentiable at \( x = 2 \), the **Mean Value Theorem is not applicable** on the interval \([0, 3]\).

Would you like more details on any of the steps?

Here are 5 related questions to explore further:

1. What is the Lagrange Mean Value Theorem and how is it applied?
2. How do you determine whether a function is continuous or differentiable?
3. Can MVT apply to functions with fractional exponents?
4. What happens when a function is not differentiable at a point in the interval?
5. How would MVT be applied if we restricted the interval to avoid the non-differentiable point?

**Tip**: When checking for the applicability of MVT, always ensure differentiability across the entire open interval, not just at the endpoints.

Is LMVT applicable for the function f(x) = (x - 2)^(1/5) over the interval [0, 3]?

Explore whether the Lagrange Mean Value Theorem (LMVT) applies to the function f(x) = (x - 2)^(1/5) on the interval [0, 3]. This problem involves concepts of continuity, differentiability, and the conditions necessary for LMVT. Detailed step-by-step analysis provided.

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