Check whether the Lagrange formula is applicable to following functions :
`f(x) =x^(2) "on" [3,4]`
`f(x) = root5 (x^(4)(x-1)) " on " [-1//2,1//2]`
If it is , find the values of `xi` appearing in this formula .

Answer

To determine whether the Lagrange Mean Value Theorem (LMVT) is applicable to the given functions, we need to check two conditions for each function:

1. The function must be continuous on the closed interval [a, b].
2. The function must be differentiable on the open interval (a, b).

Let’s analyze each function step by step.

### Function 1: \( f(x) = x^2 \) on [3, 4]

**Step 1: Check continuity on [3, 4]**
- The function \( f(x) = x^2 \) is a polynomial function.
- Polynomial functions are continuous everywhere, including the interval [3, 4].

**Step 2: Check differentiability on (3, 4)**
- The function \( f(x) = x^2 \) is also differentiable everywhere since it is a polynomial.
- Therefore, it is differentiable on the open interval (3, 4).

**Step 3: Apply the Lagrange Mean Value Theorem**
- Since both conditions are satisfied, we can apply LMVT.
- According to LMVT, there exists at least one \( c \) in (3, 4) such that:
\[ f'(c) = \frac{f(4) - f(3)}{4 - 3} \]

**Step 4: Calculate \( f(3) \) and \( f(4) \)**
- \( f(3) = 3^2 = 9 \)
- \( f(4) = 4^2 = 16 \)

**Step 5: Compute the average rate of change**
\[ \frac{f(4) - f(3)}{4 - 3} = \frac{16 - 9}{1} = 7 \]

**Step 6: Find \( f'(x) \)**
- The derivative \( f'(x) = 2x \).

**Step 7: Set \( f'(c) = 7 \)**
\[ 2c = 7 \implies c = \frac{7}{2} = 3.5 \]

**Conclusion for Function 1:**
- The LMVT is applicable, and \( c = 3.5 \) lies in the interval (3, 4).

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### Function 2: \( f(x) = \sqrt[5]{x^4(x-1)} \) on \([-1/2, 1/2]\)

**Step 1: Check continuity on \([-1/2, 1/2]\)**
- The function \( f(x) = \sqrt[5]{x^4(x-1)} \) is a composition of continuous functions (polynomial and root functions).
- The expression \( x^4(x-1) \) is continuous everywhere, and since we are taking the fifth root, it is also continuous.

**Step 2: Check differentiability on \((-1/2, 1/2)\)**
- The function is continuous, but we need to check differentiability.
- The function is not differentiable at \( x = 0 \) because the derivative involves a term that becomes undefined (the derivative of \( x^4(x-1) \) at \( x = 0 \) leads to a zero denominator in the derivative formula).

**Conclusion for Function 2:**
- The LMVT is not applicable because the function is not differentiable at \( x = 0 \).

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### Final Summary:
- For \( f(x) = x^2 \) on [3, 4]: LMVT is applicable, \( c = 3.5 \).
- For \( f(x) = \sqrt[5]{x^4(x-1)} \) on \([-1/2, 1/2]\): LMVT is not applicable.

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