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.