To solve this problem, we will first evaluate the function $f(x) = x^3 - 10x^2 + 17x + 48$ at the endpoints of the interval $[-1, 7]$, then apply the Mean Value Theorem (MVT).

Step 1: Evaluate the function at the endpoints

1. Calculate $f(-1)$: $f(-1) = (-1)^3 - 10(-1)^2 + 17(-1) + 48$ $= -1 - 10 - 17 + 48 = 20$

2. Calculate $f(7)$: $f(7) = 7^3 - 10(7^2) + 17(7) + 48$ $= 343 - 490 + 119 + 48 = 20$

Step 2: Mean Value Theorem

The Mean Value Theorem states that there exists at least one $c$ in the interval $(a, b)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$ Here, $a = -1$ and $b = 7$.

3. Calculate the average rate of change: $f'(c) = \frac{f(7) - f(-1)}{7 - (-1)} = \frac{20 - 20}{8} = 0$

Step 3: Find $c$ where $f'(c) = 0$

4. Calculate the derivative $f'(x)$: $f'(x) = 3x^2 - 20x + 17$

5. Set the derivative equal to zero: $3x^2 - 20x + 17 = 0$

Step 4: Solve the quadratic equation

Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

• Here, $a = 3$, $b = -20$, $c = 17$.

$b^2 - 4ac = (-20)^2 - 4(3)(17) = 400 - 204 = 196$ $\sqrt{196} = 14$ $x = \frac{20 \pm 14}{6}$

Calculating the two solutions:

1. $x_1 = \frac{34}{6} = \frac{17}{3} \approx 5.67$
2. $x_2 = \frac{6}{6} = 1$

Step 5: Conclusion

Thus, the values of $c$ in the interval $[-1, 7]$ that satisfy the conclusion of the Mean Value Theorem are: $c = 1 \quad \text{and} \quad c = \frac{17}{3}$

Summary

• Points for the line segment: $(-1, 20)$ and $(7, 20)$.
• Values of $c$ satisfying MVT: $1$ and $\frac{17}{3}$.

Would you like more details on any specific part of this solution, or do you have any questions? Here are some related questions to consider:

1. What is the significance of the Mean Value Theorem in calculus?
2. Can you explain how to derive a cubic function from its graph?
3. How would you find critical points of a function?
4. What is the role of the first derivative test in identifying local extrema?
5. Can you graph the function $f(x)$ along with the line segment connecting the specified points?

Tip: Always check if the endpoints yield the same value to verify applicability of the MVT!