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Let's go through the problem step by step:

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The function given is:
\[
f(x) = \frac{1}{2}x^3 - 3x^2 + 5x + 2
\]

### (a) Find the average rate of change between the points (1, 4.5) and (3.62, 4.5).

The formula for the average rate of change of a function \( f(x) \) between two points \( x_1 \) and \( x_2 \) is:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Here, \( x_1 = 1 \), \( x_2 = 3.62 \), and \( f(x_1) = f(1) = 4.5 \), \( f(x_2) = f(3.62) = 4.5 \).

So, the average rate of change:

\[
\text{Average Rate of Change} = \frac{4.5 - 4.5}{3.62 - 1} = \frac{0}{2.62} = 0
\]

Thus, the average rate of change is **0**.

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### (b) Does the Mean Value Theorem (MVT) apply to \( f \) over the interval [1, 3.62]?

The Mean Value Theorem (MVT) states that if a function is continuous and differentiable on the closed interval \([a, b]\), then there exists at least one point \( c \) in the open interval \((a, b)\) such that:

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

For the MVT to apply:
1. **Continuity**: The given function \( f(x) = \frac{1}{2}x^3 - 3x^2 + 5x + 2 \) is a polynomial, which is continuous everywhere.
2. **Differentiability**: Polynomials are also differentiable everywhere.

Thus, \( f(x) \) is both continuous and differentiable over the interval \([1, 3.62]\), so **the Mean Value Theorem applies**.

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### (c) If so, find the value(s) guaranteed by the Mean Value Theorem.

To find the value(s) of \( c \) guaranteed by the Mean Value Theorem, we need to find \( c \) such that:

\[
f'(c) = \frac{f(3.62) - f(1)}{3.62

1. Consider the function f(x) = (1/2)x^3 - 3x^2 + 5x + 2. (a) Find the average rate of change between the points (1, 4.5) and (3.62, 4.5). (b) Does the Mean Value Theorem (MVT) apply to f over the interval [1, 3.62]? (c) If so, find the value(s) guaranteed by the Mean Value Theorem.

Explore the solution to a calculus problem involving the function f(x) = (1/2)x^3 - 3x^2 + 5x + 2. Learn how to calculate the average rate of change between two points and check if the Mean Value Theorem (MVT) applies. Suitable for undergraduate calculus students.

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