Problem 46 - Calculus A New Horizon

Question

(a) Find the limits of the function as \(x \rightarrow 0^{+}\) and \(x \rightarrow+\infty .\) (b) Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$x^{2} \ln x$$

Step-by-Step Solution

Answer

Limits are \(0\) as \(x \to 0^+\) and \(+\infty\) as \(x \to \infty\). Critical point: \(x = e^{-1/2}\); inflection point: \(x = e^{-3/2}\).

Step 1: Understanding the Behavior of the Function as x Approaches Zero from the Right

To find \( \lim_{x \to 0^+} x^2 \ln x \), notice that as \( x \to 0^+ \), \( \ln x \to -\infty \). Hence, \( x^2 \ln x \) becomes \( 0 \cdot (-\infty) \), an indeterminate form. Use substitution: let \( t = \ln x \) so that \( x = e^t \) and \( x^2 = e^{2t} \). Then as \( x \to 0^+ \), \( t \to -\infty \). Rewrite the limit as \( \lim_{t \to -\infty} t e^{2t} \). Since \( e^{2t} \to 0 \) as \( t \to -\infty \), \( t e^{2t} \to 0 \), thus, \( \lim_{x \to 0^+} x^2 \ln x = 0 \).

Step 2: Finding the Limit as x Approaches Infinity

To find \( \lim_{x \to \infty} x^2 \ln x \), note that as \( x \to \infty \), \( \ln x \to \infty \). Therefore, \( x^2 \ln x \to \infty \), as \( x^2 \) grows fast enough to dominate any bounds placed by \( \ln x \). Thus, \( \lim_{x \to \infty} x^2 \ln x = +\infty \).

Step 3: Identifying Extrema and Stationary Points

To find critical points, first compute the derivative: \( f(x) = x^2 \ln x \), so \( f'(x) = 2x \ln x + x \). Set \( f'(x) = 0 \) to find critical points: \( x(2 \ln x + 1) = 0 \). The critical point comes from \( 2 \ln x + 1 = 0 \), yielding \( \ln x = -\frac{1}{2} \), or \( x = e^{-\frac{1}{2}} \).

Step 4: Identifying Inflection Points

Compute the second derivative \( f''(x) = 2 \ln x + 3 \). Set \( f''(x) = 0 \) to find possible inflection points: \( 2 \ln x + 3 = 0 \), so \( \ln x = -\frac{3}{2} \), or \( x = e^{-\frac{3}{2}} \). This points to a change in concavity at \( x = e^{-\frac{3}{2}} \).

Step 5: Graphing the Function

Using a graphing utility, plot the function \( f(x) = x^2 \ln x \). Observe the behavior as \( x \to 0^+ \), where the function touches \( y = 0 \), and as \( x \to \infty \) where the function grows infinitely. Locate and verify the relative minimum around \( x = e^{-1/2} \) and the inflection point at \( x = e^{-3/2} \) with changes in concavity on the plot.

Key Concepts

Functions

When studying functions in calculus, you are often tasked with understanding their behavior across different values of the variable. A function is a rule that assigns each input exactly one output. In our case, the function in question is \( f(x) = x^2 \ln x \). Functions are foundational elements in calculus, as they can describe real-world phenomena. Significant insights are gained by analyzing limits, such as \( \lim_{x \to 0^+} \) and \( \lim_{x \to +\infty} \). These insights reveal how functions behave near certain points, which is crucial to predict trends or outcomes in various fields.
While evaluating \( f(x) \), it's essential to understand what happens as \( x \) tends toward very small or very large values. Through this examination, we find that \( x^2 \ln x \) approaches 0 as \( x \rightarrow 0^{+} \) and \( +\infty \) as \( x \rightarrow +\infty \), showcasing the function's contrasting nature depending upon the context.

Derivatives

Derivatives are central to calculus, serving as a tool to assess the rate of change of functions. With the function \( f(x) = x^2 \ln x \), finding its derivative involves applying the product and chain rules. The derivative, \( f'(x) = 2x \ln x + x \), helps determine how fast or slow the function's value changes at specific points. This is key in identifying critical points and understanding the function's motion.
Taking derivatives can often seem challenging, but practice refines skills and makes interpreting mathematical expressions easier. The derivative provides essential data about function behavior, such as when slopes transition from increasing to decreasing, marking potential maxima, minima, or points requiring scrutiny.

Graphing Utilities

Graphing utilities, like graphing calculators or online tools, simplify visualizing intricate functions like \( f(x) = x^2 \ln x \). They are excellent for confirming analytical work done manually and provide a graphical representation to better understand function behavior.

• They allow you to plot functions swiftly, adjusting axes for precise visualization.
• These utilities often include features to calculate important points, like maxima, minima, and intercepts directly.

While analyzing \( f(x) \), employing a graphing utility reveals how the function behaves across values, how it approaches asymptotic behavior, or changes with inflection points or critical points. It's a practical complement to calculus, making complex calculations more tangible.

Critical Points

Critical points in a function are where the derivative equals zero or where the derivative does not exist. For \( f(x) = x^2 \ln x \), the critical point is found by solving \( f'(x) = 2x \ln x + x = 0 \). Simplifying this gives \( \ln x = -\frac{1}{2} \), thus \( x = e^{-\frac{1}{2}} \), identifying a potential relative extremum.
Critical points help locate where functions have peaks or troughs (relative maxima or minima). They serve as markers indicating where the function's rate of change shifts. Once found, these points must be further assessed to determine if they correspond to a maximum, minimum, or possibly a saddle point.

Inflection Points

Inflection points occur where a function's concavity changes, discernible by evaluating the second derivative. For \( f(x) = x^2 \ln x \), calculating \( f''(x) = 2 \ln x + 3 \), reveals the point where \( 2 \ln x + 3 = 0 \). This solution yields \( x = e^{-\frac{3}{2}} \), showing a shift from concave up to concave down or vice versa.
Inflection points are significant in graphically analyzing functions, indicating shifts in curvature or bending direction. Recognizing these spots reinforces the overall understanding of the function's geometrical properties and provides insights into potential regions of underestimated behavior, critical for any rigorous mathematical analysis.