Let’s analyze the given function step by step to verify and resolve part (d).

The function is:

$f(x) = \frac{1}{3}x^3 - x^2 + x - 8$

Steps to Analyze the Problem:

1. First Derivative (for critical points and intervals of increase/decrease):

$f'(x) = x^2 - 2x + 1 = (x-1)^2$

• The critical point occurs at $x = 1$ because $f'(x) = 0$.

2. Sign of $f'(x)$ (Intervals of Increase/Decrease):

• Since $f'(x) = (x-1)^2 \geq 0$, the derivative is always non-negative.
• Therefore, the function is increasing over $(-\infty, \infty)$, and there are no intervals where $f(x)$ is decreasing.

3. Second Derivative (for concavity and inflection points):

$f''(x) = 2x - 2$

4. Sign of $f''(x)$ (Concavity):

• $f''(x) = 0$ at $x = 1$.
• For $x > 1$, $f''(x) > 0$ (concave upward).
• For $x < 1$, $f''(x) < 0$ (concave downward).

Thus:

• Concave upward on $(1, \infty)$.
• Concave downward on $(-\infty, 1)$.

5. Inflection Point (where concavity changes):

An inflection point occurs when $f''(x) = 0$ and the concavity changes. Here, $f''(x) = 0$ at $x = 1$, and the concavity does change (from downward to upward).

To find the $y$-coordinate of the inflection point: $f(1) = \frac{1}{3}(1)^3 - (1)^2 + 1 - 8 = \frac{1}{3} - 1 + 1 - 8 = -\frac{23}{3}$

Thus, the inflection point is: $(x, y) = \left(1, -\frac{23}{3}\right)$

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Final Answers:

(a) Intervals of Increase/Decrease:

• Increasing: $(-\infty, \infty)$
• Decreasing: DNE

(b) Relative Extrema:

• Relative Maximum: DNE
• Relative Minimum: DNE

(c) Concavity:

• Concave Upward: $(1, \infty)$
• Concave Downward: $(-\infty, 1)$

(d) Inflection Point:

• $(x, y) = \left(1, -\frac{23}{3}\right)$

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Let me know if you need a detailed explanation for any part or assistance with similar problems!

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Related Questions:

1. How do we determine critical points from the first derivative?
2. What is the significance of an inflection point in graphing a function?
3. Why does $(x-1)^2$ indicate a flat critical point?
4. How do we interpret concavity in terms of $f''(x)$?
5. Can a function have an inflection point if $f''(x) = 0$ but concavity doesn’t change?

Tip: Always verify the concavity change when identifying inflection points!