. Check whether Lagrange's mean value theorem is applicable for each of the following

functions:

:(0,4) R defined by f(x) = |x - 1| * z

To determine whether **Lagrange's Mean Value Theorem (MVT)** is applicable to a given function on a specific interval, we need to verify the following conditions:

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

Let's now examine whether these conditions hold for the function \( f(x) = |x - 1| \cdot z \) on the interval \( (0, 4) \).

### 1. Continuity:
- The function \( f(x) = |x - 1| \) is continuous because the absolute value function is continuous everywhere.
- The multiplication by \( z \) (assuming \( z \) is constant) does not affect the continuity of \( f(x) \). So \( f(x) \) is continuous on \( (0, 4) \).

### 2. Differentiability:
- \( f(x) = |x - 1| \) is **not differentiable** at \( x = 1 \) because the absolute value function has a sharp corner at that point. Specifically:
  - For \( x > 1 \), \( |x - 1| = x - 1 \), so the function is linear and differentiable.
  - For \( x < 1 \), \( |x - 1| = 1 - x \), which is also linear but with a different slope.

  The function is not differentiable at \( x = 1 \) because the left-hand and right-hand derivatives are not equal. This lack of differentiability at \( x = 1 \) means that the function fails the second condition of Lagrange's MVT on the interval \( (0, 4) \).

### Conclusion:
Lagrange's Mean Value Theorem is **not applicable** for \( f(x) = |x - 1| \cdot z \) on the interval \( (0, 4) \) because the function is not differentiable at \( x = 1 \).

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Would you like further details on any part of this explanation or have any specific questions?

Here are 5 related questions to consider:
1. What is the geometric interpretation of Lagrange's Mean Value Theorem?
2. Can a function be continuous but not differentiable? If so, what are some examples?
3. How can we apply the Mean Value Theorem to a piecewise function?
4. What other conditions, apart from continuity and differentiability, are important for applying the MVT?
5. How does Rolle’s theorem relate to the Mean Value Theorem?

**Tip**: Differentiability implies continuity, but continuity does not always imply differentiability. Always check for points like corners or cusps where the function might not be differentiable.

Check whether Lagrange's mean value theorem is applicable for the function f(x) = |x - 1| * z on the interval (0, 4).

Explore the applicability of Lagrange's Mean Value Theorem for the function f(x) = |x - 1| * z on the interval (0, 4). This solution evaluates the function's continuity and differentiability to determine whether the theorem holds. Suitable for high school and undergraduate students studying calculus.

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