To find the point(s) $c$ where the conclusion of the Mean Value Theorem (MVT) holds for $f(x) = x^3$ on the interval $[-14, 14]$, we follow the steps of the theorem.

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Step 1: Verify that MVT applies

The MVT requires that:

1. $f(x)$ is continuous on $[a, b]$, where $a = -14$ and $b = 14$.

   • Since $f(x) = x^3$ is a polynomial, it is continuous everywhere, so it is continuous on $[-14, 14]$. ✅

2. $f(x)$ is differentiable on the open interval $(a, b) = (-14, 14)$.

   • Since $f(x) = x^3$ is differentiable everywhere (as all polynomials are differentiable), it is differentiable on $(-14, 14)$. ✅

Since both conditions are met, we can apply the Mean Value Theorem.

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Step 2: Use the Mean Value Theorem formula

The MVT states that there exists at least one $c \in (a, b)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$ Substitute $a = -14$, $b = 14$, and the function $f(x) = x^3$ into the equation.

1. Calculate $f(b)$ and $f(a)$:
   $f(14) = 14^3 = 2744, \quad f(-14) = (-14)^3 = -2744$

2. Compute the slope of the secant line:
   $\frac{f(14) - f(-14)}{14 - (-14)} = \frac{2744 - (-2744)}{28} = \frac{5488}{28} = 196$

Thus, the slope of the secant line is 196.

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Step 3: Find $c$ such that $f'(c) = 196$

1. Find $f'(x)$ for $f(x) = x^3$:
   $f'(x) = 3x^2$

2. Set $f'(c)$ equal to the slope of the secant line:
   $3c^2 = 196$

3. Solve for $c$:
   $c^2 = \frac{196}{3}$ $c = \pm \sqrt{\frac{196}{3}}$ $c = \pm \frac{\sqrt{196}}{\sqrt{3}} = \pm \frac{14}{\sqrt{3}} = \pm \frac{14 \sqrt{3}}{3}$

These are the $c$-values where the Mean Value Theorem holds.

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Final Answer

The conclusion of the Mean Value Theorem holds at the points: $c = \frac{14 \sqrt{3}}{3}, \quad c = -\frac{14 \sqrt{3}}{3}$ These are the points within the interval $(-14, 14)$ where the MVT holds.

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If you'd like more details or explanations on any step, let me know!
Here are 5 related questions you might find useful:

1. How do you know if a function is continuous and differentiable on an interval?
2. What does the slope of the secant line represent in the context of MVT?
3. How do you compute the derivative of a polynomial function?
4. Why are there two values of $c$ in this problem?
5. Can you visualize the Mean Value Theorem graphically?

Tip:
When working with the Mean Value Theorem, always check for continuity and differentiability on the given interval before applying the theorem.