determine whether the MVT applies to the function f(x) = 3x^2 - x on the interval [-1, 2]. If it applies, find al the values of c guaranteed by the MVT for the indicated interval

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 number \( c \in (a, b) \) such that:

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

### Step 1: Verify if the MVT applies
To apply the MVT, we need to check if the function \( f(x) = 3x^2 - x \) meets the conditions of the theorem:
- **Continuity**: Since \( f(x) = 3x^2 - x \) is a polynomial, it is continuous everywhere, so it is continuous on \([-1, 2]\).
- **Differentiability**: Polynomials are differentiable everywhere, so \( f(x) \) is differentiable on \((-1, 2)\).

Since both conditions are satisfied, the MVT applies.

### Step 2: Apply the Mean Value Theorem
Next, we need to find \( c \in (-1, 2) \) such that:

\[
f'(c) = \frac{f(2) - f(-1)}{2 - (-1)}
\]

#### 2.1 Calculate \( f(b) - f(a) \)
- \( f(2) = 3(2)^2 - 2 = 12 - 2 = 10 \)
- \( f(-1) = 3(-1)^2 - (-1) = 3(1) + 1 = 4 \)

Now, calculate the difference:

\[
f(2) - f(-1) = 10 - 4 = 6
\]

#### 2.2 Compute the slope of the secant line
\[
\frac{f(2) - f(-1)}{2 - (-1)} = \frac{6}{3} = 2
\]

Thus, we need to find \( c \in (-1, 2) \) such that:

\[
f'(c) = 2
\]

### Step 3: Find \( f'(x) \)
The derivative of \( f(x) = 3x^2 - x \) is:

\[
f'(x) = 6x - 1
\]

We want \( f'(c) = 2 \), so solve:

\[
6c - 1 = 2
\]

\[
6c = 3 \quad \Rightarrow \quad c = \frac{1}{2}
\]

### Conclusion:
The value of \( c \) guaranteed by the MVT is \( c = \frac{1}{2} \).

Do you have any questions or would you like more details?

Here are 5 related questions to expand on this topic:
1. Can the MVT apply to non-polynomial functions? If so, under what conditions?
2. How does the Rolle's Theorem relate to the MVT?
3. Can there be more than one value of \( c \) guaranteed by the MVT for certain functions?
4. What is the geometric interpretation of the MVT?
5. How do you verify differentiability for more complex functions?

**Tip:** When using the MVT, always check both continuity and differentiability on the appropriate intervals before proceeding with the calculation.

Determine whether the MVT applies to the function f(x) = 3x^2 - x on the interval [-1, 2]. If it applies, find all the values of c guaranteed by the MVT for the indicated interval.

Explore the application of the Mean Value Theorem (MVT) to the function f(x) = 3x^2 - x on the interval [-1, 2]. This step-by-step solution verifies the continuity and differentiability conditions and calculates the value of c that satisfies the MVT. Ideal for students in Grades 11-12.

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