The task is to graph the function $f(x) = 5x \ln(x)$ on its domain. Let's analyze and determine the domain, key points, and shape of the graph.

Step 1: Determine the Domain

The function involves $\ln(x)$, which is only defined for $x > 0$. Therefore, the domain of $f(x) = 5x \ln(x)$ is: $x > 0$

Step 2: Analyze the Function Behavior

1. Find Critical Points: To find the maximum, minimum, or points where the graph changes direction, we can take the derivative of $f(x)$ and set it to zero.
2. Intercepts: $f(x) = 5x \ln(x)$ will pass through the origin (0,0) if we consider limits approaching zero from the right. However, as $x \to 0^+$, $f(x)$ approaches negative infinity.

Step 3: Derivative of $f(x)$ to Find Critical Points

Using the product rule: $f'(x) = 5 \ln(x) + 5$ Setting $f'(x) = 0$: $5 \ln(x) + 5 = 0 \Rightarrow \ln(x) = -1 \Rightarrow x = e^{-1} = \frac{1}{e} \approx 0.3679$ This point $x = \frac{1}{e}$ is critical and will give a local minimum or maximum.

Step 4: Sketch the Graph

• The function is positive when $x > 1$.
• It has a local minimum at $x = \frac{1}{e}$.
• As $x \to \infty$, $f(x) \to \infty$.

Plotting

You can use a graphing calculator or online tool to visualize it.

Would you like more detailed steps on any part, or have any questions?

1. What are the steps to determine the range of a function like this?
2. How can derivatives help in finding the behavior of functions?
3. Why does $\ln(x)$ affect the domain of the function?
4. What would happen to the graph if the coefficient of $x \ln(x)$ was different?
5. How can limits help in understanding the function near boundaries?

Tip: Use critical points and behavior at boundaries to help sketch any complex function’s graph!