Lagrange's mean value theorem is not applicable to f(x) in [1,4] where f(x) =

Options

A. `x^(2) -2x`
B. |x-2|
C. x|x|
D. `x^(3)`

Answer

To determine which function does not satisfy Lagrange's Mean Value Theorem (LMVT) in the interval [1, 4], we need to check the two main conditions of the theorem:

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 analyze each option step by step.

### Step-by-Step Solution:

**Option 1: \( f(x) = x^2 - 2x \)**

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

2. **Check Differentiability:**
- The derivative \( f'(x) = 2x - 2 \) exists for all \( x \).
- Therefore, \( f(x) \) is differentiable on (1, 4).

**Conclusion for Option 1:** Satisfies both conditions.

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**Option 2: \( f(x) = |x - 2| \)**

1. **Check Continuity:**
- The function \( f(x) = |x - 2| \) is continuous everywhere, including the interval [1, 4].

2. **Check Differentiability:**
- The function has a sharp corner at \( x = 2 \).
- The derivative does not exist at \( x = 2 \) because the left-hand derivative and right-hand derivative are not equal.

**Conclusion for Option 2:** Satisfies continuity but not differentiability.

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**Option 3: \( f(x) = x \cdot |x| \)**

1. **Check Continuity:**
- For \( x \in [1, 4] \), \( |x| = x \).
- Thus, \( f(x) = x^2 \), which is continuous.

2. **Check Differentiability:**
- The derivative \( f'(x) = 2x \) exists for all \( x \).
- Therefore, \( f(x) \) is differentiable on (1, 4).

**Conclusion for Option 3:** Satisfies both conditions.

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**Option 4: \( f(x) = x^3 \)**

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

2. **Check Differentiability:**
- The derivative \( f'(x) = 3x^2 \) exists for all \( x \).
- Therefore, \( f(x) \) is differentiable on (1, 4).

**Conclusion for Option 4:** Satisfies both conditions.

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### Final Conclusion:
The function that does not satisfy Lagrange's Mean Value Theorem in the interval [1, 4] is **Option 2: \( f(x) = |x - 2| \)**, as it is not differentiable at \( x = 2 \).