Problem 38 - Essential Calculus

Question

Find all relative extrema. Use the Second Derivative Test where applicable. \(y=x^{2} \log _{3} x\)

Step-by-Step Solution

Answer

The function \(y=x^{2} \log _{3} x\) has a relative maximum at x= \(3^{-0.5}\). The test is inconclusive for x=0.

Step 1: Find the derivative

The first step is to find the first derivative of the function. Since \(y=x^{2} \log _{3} x\), we have to use both the power rule and the product rule of derivatives. The derivative would be \(y'=x\)(2\(\log_{3}x\)+1).

Step 2: Set the derivative equal to zero and solve for x

Secondly, we need to set the derivative equal to zero to find the critical points. Solving the equation \(y'=x\)(2\(\log_{3}x\)+1) = 0, we get x=0 and x=\(3^{-0.5}\).

Step 3: Apply second derivative test

To determine if the critical points are relative minimum or maximum, we can apply the second derivative test. The second derivative of \(y\)=\(x^{2} \log _{3} x\) is \(y'' = 2log_{3}x + (2/x)\). Evaluating \(y''\) at x= \(3^{-0.5}\) gives a negative result, implying a relative maximum. Evaluating at x=0 is undefined so the test isn't conclusive.

Step 4: Interpret the results

Since the second derivative at x=\(3^{-0.5}\) results in a negative value, there is a relative maximum at that point as per the Second Derivative Test. The test isn't conclusive for x=0 so further tests (like First derivative test) or simply analyzing the graph would be helpful.

Key Concepts

Relative Extrema

Relative extrema refer to points on the graph of a function where the function takes a relative minimum or maximum value. These are the peaks and troughs of a graph, which can help us understand the overall behavior of the function.
To find these points, we look for where the derivative of the function equals zero or does not exist. These are known as critical points. A relative maximum is a point where the function changes from increasing to decreasing, and a relative minimum is a point where the function changes from decreasing to increasing.
When finding extrema for functions such as polynomial or logarithmic products, we first compute the derivatives and evaluate changes in sign. Once the critical points are determined, they can be classified using tests like the First Derivative Test or the Second Derivative Test.

Product Rule

The product rule is a fundamental principle in calculus used to differentiate functions that are multiplied together. Suppose we have a function that can be expressed as the product of two functions, such as \(y = u(x) \cdot v(x)\). The product rule helps in finding the derivative of this composite function.
The product rule states that the derivative of the product of two functions is given by the formula:

• \( (uv)' = u'v + uv' \)

In practice, you differentiate the first function, keep the second function as is, and then add the product of keeping the first function as is while differentiating the second. Applying the product rule ensures you capture all necessary changes in the function when both elements contribute to the slope or rate of change.

Power Rule

One of the most straightforward and essential rules for differentiation in calculus is the power rule. It provides a quick way to find derivatives for functions of the form \(f(x) = x^n\), where \(n\) is any real number.
The power rule states that:

• \((x^n)' = nx^{n-1}\)

The process involves multiplying the current exponent by the base and reducing the exponent by one. For instance, for a simple function like \(x^2\), the derivative using the power rule is \(2x^{2-1} = 2x\). This rule becomes especially powerful when used in combination with other rules, such as the product rule, where multiple terms within a function need differentiation.

Second Derivative Test

The second derivative test is a useful method to identify the nature of critical points found with the first derivative of a function. By calculating the second derivative, or \(f''(x)\), we can determine whether each critical point is a relative maximum, a relative minimum, or neither.
For any critical point \(c\):

• If \(f''(c) > 0\), the function \(f\) has a relative minimum at \(c\).
• If \(f''(c) < 0\), the function \(f\) has a relative maximum at \(c\).
• If \(f''(c) = 0\), the test is inconclusive, so other methods such as graphing might be necessary.

In practical terms, the second derivative reflects the curvature of the graph. A positive second derivative indicates concavity upwards, suggesting a trough or minimum, whereas a negative second derivative shows concavity downwards, indicating a peak or maximum. This test helps clarify the behavior and shape of the graph around critical points.