The **Lagrange Mean Value Theorem (LMVT)**, also known as the **Mean Value Theorem (MVT)**, states that if a function \( f(x) \) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \( c \in (a, b) \) such that:

\[
f'(c) = \frac{f(b) - f(a)}{b - a}
\]

### To apply LMVT, two conditions must be satisfied:
1. **Continuity** on the closed interval \([a, b]\).
2. **Differentiability** on the open interval \((a, b)\).

### Now, let's analyze the function \( F(x) = |x| \) on the interval \([0, 1]\):

1. **Continuity**:  
   The absolute value function \( F(x) = |x| \) is continuous on the interval \([0, 1]\) because the absolute value function is continuous everywhere.

2. **Differentiability**:  
   The function \( F(x) = |x| \) is differentiable on the open interval \((0, 1)\), but **not differentiable at \( x = 0 \)**. At \( x = 0 \), there is a sharp corner in the graph of \( F(x) \), so the derivative does not exist at that point.

### Conclusion:
Since \( F(x) = |x| \) is not differentiable at \( x = 0 \), we **cannot apply LMVT** on the interval \([0, 1]\) because one of the conditions (differentiability on the entire open interval) is not met.

Do you need more details or have any questions?

Here are 5 related questions you might find interesting:
1. What happens if we apply LMVT on an interval where the function is not differentiable?
2. Can the Mean Value Theorem be applied on non-differentiable functions at points?
3. How is LMVT different from Rolle’s Theorem?
4. Can LMVT be applied on the absolute value function on a different interval?
5. What are the implications of non-differentiability on the existence of a tangent line?

**Tip:** When checking if you can apply LMVT, always verify the differentiability of the function across the entire open interval, not just continuity.

Explore whether the Lagrange Mean Value Theorem (LMVT) can be applied to the function F(x) = |x| on the interval [0,1]. This problem delves into key calculus concepts such as continuity, differentiability, and absolute value functions, highlighting the conditions needed for applying LMVT.

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