75. b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
g(x) = x(ln x)²

Verified step by step guidance

1. First, identify the domain of the function \(g(x) = x(\ln x)^2\). Since \(\ln x\) is defined only for \(x > 0\), the domain is \(x > 0\).
2. Next, find the first derivative \(g'(x)\) using the product rule. Let \(u = x\) and \(v = (\ln x)^2\). Then, \(g'(x) = u'v + uv'\). Calculate \(u' = 1\) and \(v' = 2 \ln x \cdot \frac{1}{x} = \frac{2 \ln x}{x}\), so \(g'(x) = (1)(\ln x)^2 + x \cdot \frac{2 \ln x}{x} = (\ln x)^2 + 2 \ln x\).
3. Set the first derivative equal to zero to find critical points: \((\ln x)^2 + 2 \ln x = 0\). Factor this as \(\ln x (\ln x + 2) = 0\), which gives \(\ln x = 0\) or \(\ln x = -2\). Solve for \(x\) to find critical points: \(x = e^0 = 1\) and \(x = e^{-2}\).
4. Use the second derivative test to classify each critical point. Find \(g''(x)\) by differentiating \(g'(x) = (\ln x)^2 + 2 \ln x\). Differentiate term-by-term: \(\frac{d}{dx} (\ln x)^2 = 2 \ln x \cdot \frac{1}{x} = \frac{2 \ln x}{x}\) and \(\frac{d}{dx} (2 \ln x) = \frac{2}{x}\). So, \(g''(x) = \frac{2 \ln x}{x} + \frac{2}{x} = \frac{2(\ln x + 1)}{x}\).
5. Evaluate \(g''(x)\) at each critical point to determine concavity: if \(g''(x) > 0\), the point is a local minimum; if \(g''(x) < 0\), it is a local maximum. Also, consider the behavior of \(g(x)\) as \(x \to 0^+\) and as \(x \to \infty\) to identify any absolute extrema on the domain.

Key Concepts

Critical Points and Derivatives

Critical points occur where the derivative of a function is zero or undefined. Finding these points helps identify potential local maxima, minima, or saddle points by analyzing where the slope of the tangent line changes.

Local and Absolute Extrema

Local extrema are points where a function reaches a maximum or minimum within a small neighborhood, while absolute extrema are the highest or lowest values over the entire domain. Determining these requires evaluating the function at critical points and boundaries.

Domain Considerations for Logarithmic Functions

Since g(x) = x(ln x)² involves ln(x), the domain is restricted to x > 0. Understanding this domain is essential to correctly identify where extrema can occur and to avoid invalid points outside the domain.