To solve the problem given in the image, let's address each part in turn.

Q1: Find the Relative Extreme of the Function $g(x) = 3 - x^{1/3}$

To find the relative extrema of the function $g(x) = 3 - x^{1/3}$, we need to:

1. Find the first derivative $g'(x)$:
   The first derivative will help us determine where the slope of the function is zero or undefined, which corresponds to potential relative extrema.

   $g(x) = 3 - x^{1/3}$ Taking the derivative with respect to $x$:

   $g'(x) = 0 - \frac{1}{3}x^{-2/3} = -\frac{1}{3}x^{-2/3} = -\frac{1}{3x^{2/3}}$

2. Set $g'(x) = 0$ to find critical points:
   Since $g'(x) = -\frac{1}{3x^{2/3}}$, setting this equal to zero does not yield any real solutions because the function is undefined at $x = 0$. This means there are no points where the derivative is zero, but we still need to check the behavior around $x = 0$.

3. Analyze the sign of $g'(x)$:

   • For $x > 0$, $g'(x)$ is negative, which means the function is decreasing.
   • For $x < 0$, $g'(x)$ is positive, which means the function is increasing.

   At $x = 0$, the derivative is undefined, but the function changes from increasing to decreasing, indicating a cusp point. Therefore, $x = 0$ is a relative maximum.

Q2: Sketch the Graph

Given the analysis above:

• As $x \to -\infty$, $g(x)$ increases without bound.
• As $x \to \infty$, $g(x)$ decreases towards negative infinity.
• There is a relative maximum at $x = 0$, and $g(0) = 3$.

The graph of $g(x) = 3 - x^{1/3}$ will have a cusp at $x = 0$ with a maximum at $(0, 3)$. The function is increasing on the left of $x = 0$ and decreasing on the right.

Would you like a more detailed explanation, or do you have any specific questions?

Here are some related questions you might find helpful:

1. How can you determine whether a function has a cusp, corner, or inflection point at a critical point?
2. What are the implications of having a derivative that is undefined at a certain point?
3. How does the behavior of $g(x)$ as $x$ approaches positive and negative infinity influence the graph?
4. Can you find the second derivative of $g(x)$ and use it to confirm the concavity?
5. How do changes in the function $g(x) = 3 - x^{1/3}$ affect its relative extrema?

Tip: Understanding the behavior of the first derivative is key to identifying and classifying relative extrema, cusps, and inflection points.