We are given the function $f(x) = 3 + x^{\frac{2}{3}}$. To find the relative extreme points, we need to follow these steps:

Step 1: Find the first derivative

To locate the relative extrema, we need the first derivative $f'(x)$.

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

Differentiate $f(x)$ with respect to $x$:

$f'(x) = \frac{d}{dx} \left( 3 + x^{\frac{2}{3}} \right) = \frac{2}{3}x^{-\frac{1}{3}} = \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}}$

So, the first derivative is:

$f'(x) = \frac{2}{3x^{\frac{1}{3}}}$

Step 2: Set the first derivative equal to zero

To find critical points, set $f'(x) = 0$ and solve for $x$.

$\frac{2}{3x^{\frac{1}{3}}} = 0$

Since the fraction $\frac{2}{3x^{\frac{1}{3}}}$ can never equal zero (there is no value of $x$ that makes this zero), there are no critical points where $f'(x) = 0$.

Step 3: Check for points where the derivative does not exist

Next, we check where the derivative does not exist. The derivative $f'(x) = \frac{2}{3x^{\frac{1}{3}}}$ is undefined when $x = 0$, since division by zero is undefined. So, $x = 0$ is a point where the derivative does not exist.

Step 4: Determine the behavior at $x = 0$

To understand the behavior of the function at $x = 0$, we can examine the behavior of $f(x)$ as $x$ approaches zero from both the positive and negative sides.

• As $x \to 0^+$ (from the right), $f'(x) = \frac{2}{3x^{\frac{1}{3}}}$ becomes large and positive, indicating that $f(x)$ is increasing.
• As $x \to 0^-$ (from the left), $f'(x) = \frac{2}{3x^{\frac{1}{3}}}$ becomes large and negative, indicating that $f(x)$ is decreasing.

Thus, there is a cusp at $x = 0$, and the function has a relative minimum at $x = 0$.

Step 5: Conclusion on relative extrema

• The function has a relative minimum at $x = 0$.
• There are no relative maxima.

Graph of the function

The graph of $f(x) = 3 + x^{\frac{2}{3}}$ is a smooth curve with a cusp at $x = 0$. The function decreases as $x$ approaches 0 from the left and increases as $x$ moves away from 0 to the right. At $x = 0$, the function reaches its minimum value of $f(0) = 3$.

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Would you like me to generate the sketch of the graph for you?

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Here are five related questions:

1. How do you determine if a function has a cusp or a smooth minimum?
2. Can the first derivative test always identify extreme points for all functions?
3. What is the significance of the point where the derivative does not exist?
4. How does the behavior of $f(x) = x^{\frac{2}{3}}$ differ from $f(x) = x^2$ near the origin?
5. How do you interpret the concavity of the function near $x = 0$?

Tip: For any function, always check where the derivative does not exist, as this might indicate cusps, corners, or vertical tangents.