In Exercises 115 and 116, find the absolute maximum and minimum values of each function on the given interval.
116. y = 10x (2 - ln(x)), (0, e²]"133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed

Verified step by step guidance

1. First, rewrite the function to a more convenient form. Given \( f(x) = x^2 \ln\left(\frac{1}{x}\right) \), use the logarithm property \( \ln\left(\frac{1}{x}\right) = -\ln(x) \) to rewrite it as \( f(x) = x^2 (-\ln(x)) = -x^2 \ln(x) \).
2. Next, find the critical points by computing the derivative \( f'(x) \). Use the product rule on \( -x^2 \ln(x) \): \( f'(x) = -\left( 2x \ln(x) + x^2 \cdot \frac{1}{x} \right) = -\left( 2x \ln(x) + x \right) \).
3. Set the derivative equal to zero to find critical points: \( f'(x) = 0 \Rightarrow -\left( 2x \ln(x) + x \right) = 0 \Rightarrow 2x \ln(x) + x = 0 \). Factor out \( x \): \( x(2 \ln(x) + 1) = 0 \). Since \( x > 0 \), solve \( 2 \ln(x) + 1 = 0 \) for \( x \).
4. Solve for \( x \) from the equation \( 2 \ln(x) + 1 = 0 \): \( \ln(x) = -\frac{1}{2} \), so \( x = e^{-1/2} \). This is the critical point inside the domain.
5. Evaluate \( f(x) \) at the critical point \( x = e^{-1/2} \) and at the endpoints of the domain (if given) to determine the absolute maximum value and where it is assumed.

Key Concepts

Finding Critical Points

Critical points occur where the derivative of a function is zero or undefined. To find absolute extrema on a closed interval, first compute the derivative, set it equal to zero, and solve for x-values within the interval. These points are candidates for local maxima or minima.

Evaluating Function at Endpoints

Absolute maximum and minimum values on a closed interval can occur at critical points or at the interval's endpoints. After finding critical points, evaluate the function at these points and at the endpoints to determine the absolute extrema.

Properties of Logarithmic Functions in Calculus

Logarithmic functions like ln(x) have specific domains and behaviors that affect differentiation and evaluation. Understanding how to differentiate functions involving ln(x) and handle expressions like ln(1/x) is essential for correctly finding derivatives and analyzing function behavior.