https://file.aiquickdraw.com/mathbot/file/17305931587881gmcw9o7.jpg

To solve this problem, let's apply the Mean Value Theorem (MVT). The 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 \( c \) in \((a, b)\) such that:

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

### Step 1: Identify the Function and Interval

The given function is:
\[
f(x) = 9x + \frac{9}{x}
\]
The interval is \(\left[\frac{1}{19}, 19\right]\), where \( a = \frac{1}{19} \) and \( b = 19 \).

### Step 2: Compute \( f(a) \) and \( f(b) \)

1. **Calculate \( f\left(\frac{1}{19}\right) \):**
   \[
   f\left(\frac{1}{19}\right) = 9 \cdot \frac{1}{19} + \frac{9}{\frac{1}{19}} = \frac{9}{19} + 9 \cdot 19 = \frac{9}{19} + 171
   \]

2. **Calculate \( f(19) \):**
   \[
   f(19) = 9 \cdot 19 + \frac{9}{19} = 171 + \frac{9}{19}
   \]

### Step 3: Compute the Average Rate of Change \(\frac{f(b) - f(a)}{b - a}\)

The average rate of change of \( f(x) \) on \(\left[\frac{1}{19}, 19\right]\) is:
\[
\frac{f(b) - f(a)}{b - a} = \frac{\left(171 + \frac{9}{19}\right) - \left(\frac{9}{19} + 171\right)}{19 - \frac{1}{19}} = \frac{0}{19 - \frac{1}{19}}
\]

So

Find the value or values of c that satisfy the equation f(b) - f(a) / b - a = f'(c) in the conclusion of the Mean Value Theorem for the given function and interval, where f(x) = 9x + 9/x and the interval is [1/19, 19].

Solve for the value of c that satisfies the Mean Value Theorem for the function f(x) = 9x + 9/x on the interval [1/19, 19]. This calculus problem involves calculating derivatives and applying the Mean Value Theorem to find the average rate of change over a specified interval.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation