Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = x³ ln x on (0, ∞)

Verified step by step guidance

1. To find the critical points of the function \( f(x) = x^3 \ln x \) on the interval \( (0, \infty) \), first find the derivative \( f'(x) \). Use the product rule: if \( u(x) = x^3 \) and \( v(x) = \ln x \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \).
2. Calculate the derivatives: \( u'(x) = 3x^2 \) and \( v'(x) = \frac{1}{x} \). Substitute these into the product rule to get \( f'(x) = 3x^2 \ln x + x^3 \cdot \frac{1}{x} \). Simplify this expression.
3. Set \( f'(x) = 0 \) to find the critical points. This gives the equation \( 3x^2 \ln x + x^2 = 0 \). Factor out \( x^2 \) to get \( x^2(3 \ln x + 1) = 0 \). Since \( x^2 \neq 0 \) for \( x > 0 \), solve \( 3 \ln x + 1 = 0 \).
4. Solve \( 3 \ln x + 1 = 0 \) to find the critical points. This simplifies to \( \ln x = -\frac{1}{3} \). Exponentiate both sides to solve for \( x \), giving \( x = e^{-\frac{1}{3}} \).
5. To identify the absolute maximum and minimum values, evaluate \( f(x) \) at the critical point \( x = e^{-\frac{1}{3}} \) and consider the behavior of \( f(x) \) as \( x \to 0^+ \) and \( x \to \infty \). Compare these values to determine the absolute extrema on the interval \( (0, \infty) \).

Key Concepts

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local maxima and minima, as they indicate where the function's slope changes. To find critical points, one must first compute the derivative of the function and solve for the values of x that satisfy these conditions.

Absolute Maximum and Minimum

The absolute maximum and minimum values of a function on a given interval are the highest and lowest values that the function attains within that interval. To determine these values, one must evaluate the function at its critical points and at the endpoints of the interval, comparing these values to find the overall maximum and minimum.

Natural Logarithm Function

The natural logarithm function, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.718. It is defined for positive x and plays a crucial role in calculus, particularly in functions involving growth and decay. Understanding its properties, such as its domain and behavior as x approaches zero or infinity, is vital for analyzing functions like ƒ(x) = x³ ln x.