Question

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

Short 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-by-step solution

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).

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=\(3^{-0.5}\).

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.

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.