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The problem asks to find the value or values of \( c \) that satisfy the equation

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

for the given function and interval, using the Mean Value Theorem (MVT).

### Step 1: Analyze the Given Function and Interval
The function is:
\[
f(x) = 7x + \frac{7}{x}
\]
And the interval is \(\left[\frac{1}{10}, 10\right]\).

### Step 2: Mean Value Theorem
According to the Mean Value Theorem, there exists at least one value \( c \) in the open interval \( \left( \frac{1}{10}, 10 \right) \) such that:

\[
f'(c) = \frac{f(10) - f\left(\frac{1}{10}\right)}{10 - \frac{1}{10}}
\]

### Step 3: Compute \( f'(x) \)
To apply the MVT, we first need the derivative \( f'(x) \).

\[
f(x) = 7x + \frac{7}{x}
\]
Using the power rule and the chain rule, we differentiate:

\[
f'(x) = 7 - \frac{7}{x^2}
\]

### Step 4: Calculate \( f(10) \) and \( f\left(\frac{1}{10}\right) \)

- \( f(10) = 7(10) + \frac{7}{10} = 70 + 0.7 = 70.7 \)
- \( f\left(\frac{1}{10}\right) = 7\left(\frac{1}{10}\right) + \frac{7}{\frac{1}{10}} = 0.7 + 70 = 70.7 \)

### Step 5: Compute the Difference Quotient

Now calculate:

\[
\frac{f(10) - f\left(\frac{1}{10}\right)}{10 - \frac{1}{10}} = \frac{70.7 - 70.7}{10 - 0.1} = \frac{0}{9.9} = 0
\]

Thus, the right-hand side of the Mean Value Theorem equation is \( 0 \).

### Step 6: Set \( f'(c) = 0 \)

From the MVT, we now have:

\[
7 - \frac{7}{c^2} = 0
\]

Solve for \( c \):

\[
7 = \frac{7}{c^2}
\]

\[
c^2 = 1
\]

\[
c = \pm 1
\]

### Step 7: Check If \( c \) Lies in the Interval
The interval is \( \left(\frac{1}{10}, 10\right) \). So, \( c = 1 \) is the only valid solution because \( c = -1 \) is not in the interval.

### Final Answer:
The value of \( c \) that satisfies the Mean Value Theorem is:

\[
c = 1
\]

---

Would you like more details or have any other questions?

Here are 5 relative questions for further exploration:
1. What happens if the function was different, such as \( f(x) = x^3 \)?
2. How would the problem change if the interval were \( [2, 5] \)?
3. Why does the MVT require a differentiable function on an open interval?
4. How is the Mean Value Theorem applied in real-world scenarios?
5. What is the difference between the Mean Value Theorem and Rolle's Theorem?

**Tip**: The Mean Value Theorem guarantees at least one value \( c \), but in some cases, multiple \( c \)-values might satisfy the equation depending on the function's behavior.

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. f(x) = 7x + 7/x on the interval [1/10, 10].

Solve a calculus problem using the Mean Value Theorem for the function f(x) = 7x + 7/x on the interval [1/10, 10]. The problem involves finding the value of c such that the average rate of change equals the instantaneous rate of change. A detailed solution includes calculating the derivative and solving for c.

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