please show all step by step work. thank you! ### Graphing \( f(x) = x \ln(x) \)Task: Sketch the graph of \( f(x) = x \ln(x) \) and label important points on the graph.Guidelines:- Look for intercepts, domain, range, critical points, and inflection points.- Consider asymptotes as well.**Tip:** You may need to use L'Hôpital's Rule to analyze the function.### Step-by-Step Instructions1. **Intercepts:** - To find the intercepts, solve \( f(x) = 0 \). - Since \( f(x) = x \ln(x) \), set \( x \ln(x) = 0 \). - \( x = 0 \) or \( \ln(x) = 0 \). - \( \ln(x) = 0 \) implies \( x = 1 \). - Therefore, the x-intercepts are at \( x = 0 \) and \( x = 1 \).2. **Domain:** - The natural logarithm function, \( \ln(x) \), is defined for \( x > 0 \). - Hence, the domain of \( f(x) \) is \( (0, \infty) \).3. **Range:** - To find the range of \( f(x) \), analyze the behavior as \( x \to 0^+ \) and as \( x \to \infty \). - As \( x \to 0^+ \), \( x \ln(x) \to 0 \) (approaching from below the x-axis). - As \( x \to \infty \), \( x \ln(x) \to \infty \). - Thus, the range of \( f(x) \) is \( (-\infty, \infty) \).4. **Critical Points:** - To find critical points, compute the derivative \( f'(x) \). - \( f'(x) = \frac{d}{dx}[x \ln(x)] = \ln(x) + 1 \). - Set the derivative to zero and solve for \( x \). - \( \ln(x) + 1 = 0 \) implies \( \ln(x) = -1 \). - Solving for \( x \), \( x

Name: please show all step by step work. thank you! ### Graphing \( f(x) = x
Uploaded: 2024-03-05T11:53:49.000Z
Channel: Bartleby
Description: please show all step by step work. thank you! ### Graphing \( f(x) = x \ln(x) \)Task: Sketch the graph of \( f(x) = x \ln(x) \) and label important points on the graph.Guidelines:- Look for intercepts, domain, range, critical points, and inflection p