d:10,3]→R defined by /(x)=(0, x < 1 x >= 1 lmvt is applicable 

It looks like you have a piecewise function \( f(x) \) defined on the interval \( (1, 3) \to \mathbb{R} \), and you want to determine whether the **Lagrange Mean Value Theorem (LMVT)** is applicable.

For LMVT to apply to a function \( f \) on an interval \( [a, b] \), the following conditions must be met:

1. \( f(x) \) must be **continuous** on \( [a, b] \).
2. \( f(x) \) must be **differentiable** on \( (a, b) \).

Here, let's assume your piecewise function \( f(x) \) is defined as:
\[
f(x) = 
\begin{cases} 
0, & \text{if } x < 1 \\
x, & \text{if } x \geq 1
\end{cases}
\]

### 1. Checking continuity on the interval \( [1, 3] \):
- The function \( f(x) = x \) for \( x \geq 1 \) is continuous at \( x = 1 \) because:
  \[
  \lim_{x \to 1^-} f(x) = 0 \quad \text{and} \quad \lim_{x \to 1^+} f(x) = 1
  \]
  Since the limits from the left and right do not match, the function is **discontinuous** at \( x = 1 \).

### 2. Checking differentiability on \( (1, 3) \):
- For \( x > 1 \), the function \( f(x) = x \) is differentiable, as its derivative is simply \( f'(x) = 1 \).
- However, because the function is not continuous at \( x = 1 \), it cannot be differentiable at \( x = 1 \).

### Conclusion:
Since the function is not continuous at \( x = 1 \), **LMVT is not applicable** to this function on the interval \( [1, 3] \).

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. What are the necessary conditions for LMVT?
2. Can LMVT be applied to a function with a jump discontinuity?
3. What is the difference between LMVT and Rolle’s Theorem?
4. How would the theorem apply if the function were modified to be continuous at \( x = 1 \)?
5. How do you prove a function is differentiable?

**Tip**: Continuity is crucial for applying most calculus theorems, including LMVT, so always check for it before verifying differentiability!

d:10,3]→R defined by /(x)=(0, x < 1 x >= 1 lmvt is applicable

Explore whether the Lagrange Mean Value Theorem (LMVT) applies to the piecewise function defined as f(x) = 0 for x < 1 and f(x) = x for x >= 1 on the interval [1, 3]. This problem covers concepts like continuity, differentiability, and the conditions for applying LMVT. Suitable for students in Grades 11-12.

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