To find the relative extrema of the function $f(x) = \frac{\ln(x^2)}{x}$ using the second derivative test, let's go through the following steps:

1. Rewrite the Function: $f(x) = \frac{\ln(x^2)}{x} = \frac{2\ln(x)}{x}, \quad x \neq 0$

2. Find the First Derivative $f'(x)$: Using the quotient rule $\left( \frac{g}{h} \right)' = \frac{g'h - gh'}{h^2}$:

   • $g(x) = 2 \ln(x)$ and $h(x) = x$
   • $g'(x) = \frac{2}{x}$ and $h'(x) = 1$

   So, $f'(x) = \frac{\left( \frac{2}{x} \right) x - 2\ln(x) \cdot 1}{x^2} = \frac{2 - 2\ln(x)}{x^2} = \frac{2(1 - \ln(x))}{x^2}$

3. Set the First Derivative Equal to Zero: To find the critical points, solve $f'(x) = 0$: $\frac{2(1 - \ln(x))}{x^2} = 0$ This implies $1 - \ln(x) = 0$ or $\ln(x) = 1$, so $x = e$.

4. Find the Second Derivative $f''(x)$: Now, we need to find $f''(x)$ to apply the second derivative test.

   Differentiating $f'(x) = \frac{2(1 - \ln(x))}{x^2}$ with respect to $x$:

   • Use the quotient rule again: $f''(x) = \frac{\left(-\frac{2}{x}\right) x^2 - 2(1 - \ln(x)) \cdot 2x}{x^4} = \frac{-2x^2 - 4x(1 - \ln(x))}{x^4}$ Simplifying, we get: $f''(x) = \frac{-2x - 4(1 - \ln(x))}{x^3} = \frac{-2x + 4\ln(x) - 4}{x^3} = \frac{4\ln(x) - 2x - 4}{x^3}$

5. Evaluate $f''(x)$ at the Critical Point $x = e$: Substitute $x = e$ into $f''(x)$: $f''(e) = \frac{4\ln(e) - 2e - 4}{e^3} = \frac{4 \cdot 1 - 2e - 4}{e^3} = \frac{-2e}{e^3} = \frac{-2}{e^2}$ Since $f''(e) < 0$, there is a relative maximum at $x = e$.

6. Check for Any Additional Critical Points or End Behavior: Given the nature of $f(x)$, it only has one critical point at $x = e$, and by the second derivative test, it is a relative maximum.

Thus, the function $f(x) = \frac{\ln(x^2)}{x}$ has a relative maximum at $x = e$.

Would you like a more detailed breakdown of each step or additional insights?

Related Questions

1. How would you find absolute extrema for this function over a closed interval?
2. Can you analyze the end behavior of this function as $x \to 0^+$ and $x \to \infty$?
3. What does the function's graph look like around the relative extrema?
4. How do logarithmic and rational functions affect extrema in general?
5. Could you determine the concavity intervals of $f(x)$ based on the second derivative?

Tip:

Remember that the second derivative test is only conclusive when $f''(c) \neq 0$ at critical points. If $f''(c) = 0$, the test is inconclusive, and other methods may be required.