Let's work through each part of the problem with the information given:

Given Information:

• $f'(x) = \frac{2}{3\sqrt[3]{x}} - 2$
• $f''(x) = -\frac{2}{3x^{4/3}}$

Part (a): Determine Intervals of Increase and Decrease

To find where $f(x)$ is increasing or decreasing, analyze the sign of $f'(x)$.

1. Find Critical Points of $f'(x)$: Set $f'(x) = 0$: $\frac{2}{3\sqrt[3]{x}} - 2 = 0$ Simplify by isolating $\frac{2}{3\sqrt[3]{x}}$: $\frac{2}{3\sqrt[3]{x}} = 2$ Multiply both sides by $3\sqrt[3]{x}$: $2 = 6\sqrt[3]{x}$ Divide by 6: $\sqrt[3]{x} = \frac{1}{3}$ Cube both sides: $x = \frac{1}{27}$ So, $x = \frac{1}{27}$ is a critical point.

2. Determine the Sign of $f'(x)$ Around $x = \frac{1}{27}$:

   • For $x < \frac{1}{27}$, plug a test point (e.g., $x = 0.001$) into $f'(x)$ and determine if it's positive or negative.
   • For $x > \frac{1}{27}$, plug a test point (e.g., $x = 1$) into $f'(x)$.

   After testing values:

   • If $f'(x) > 0$, $f(x)$ is increasing.
   • If $f'(x) < 0$, $f(x)$ is decreasing.

3. Identify Local Maxima or Minima: If $f(x)$ changes from increasing to decreasing at $x = \frac{1}{27}$, it would be a local maximum. If it changes from decreasing to increasing, it would be a local minimum.

Part (b): Determine Concavity and Inflection Points

To find where $f(x)$ is concave up or down, analyze the sign of $f''(x)$.

1. Examine $f''(x)$: $f''(x) = -\frac{2}{3x^{4/3}}$.

   • Notice that $f''(x)$ is always negative for $x > 0$.
   • Therefore, $f(x)$ is concave down on $(0, \infty)$.

2. Check for Inflection Points: Since $f''(x)$ does not change sign (it's negative for all $x > 0$), there are no inflection points in this domain.

Part (c): Graph the Function

Without the explicit form of $f(x)$, we can only approximate its behavior using the information from the derivatives:

• Increasing/Decreasing Behavior: $f(x)$ will increase on the interval determined in part (a) and decrease on the other.
• Concavity: $f(x)$ will be concave down for all $x > 0$.
• Critical Points: Plot the critical point $x = \frac{1}{27}$ to illustrate a maximum or minimum if identified in part (a).

Would you like detailed steps for part (a) or (b), or do you have any questions about any specific part?

Follow-up Questions:

1. What is the importance of determining intervals of increase and decrease for a function?
2. How does the second derivative help in understanding the concavity of a function?
3. Why are inflection points significant in graphing functions?
4. How can you approximate the shape of $f(x)$ even without knowing its exact equation?
5. How do critical points affect the behavior of a function?

Tip:

Remember, inflection points occur where the second derivative changes sign, indicating a change in concavity.