To find the relative extrema of the function $y = \ln(x^3 + 3x^2 + 5)$, we need to:

1. Find the derivative of $y$ with respect to $x$.
2. Set the derivative equal to zero to find critical points.
3. Determine the nature of these critical points (relative minimum, maximum, or neither) using the second derivative or other methods.

Step 1: Derivative of $y$

Given: $y = \ln(x^3 + 3x^2 + 5)$

Using the chain rule, the derivative $y'$ is: $y' = \frac{1}{x^3 + 3x^2 + 5} \cdot (3x^2 + 6x) = \frac{3x^2 + 6x}{x^3 + 3x^2 + 5}$

We can simplify this to: $y' = \frac{3x(x + 2)}{x^3 + 3x^2 + 5}$

Step 2: Set the Derivative to Zero

To find critical points, set $y' = 0$: $\frac{3x(x + 2)}{x^3 + 3x^2 + 5} = 0$

This implies: $3x(x + 2) = 0$

So, $x = 0$ or $x = -2$.

Step 3: Determine the Nature of Critical Points

To identify whether these points are maxima, minima, or neither, we can use the second derivative test or examine the behavior of $y'$ around these points.

Check for $x = 0$:

1. Substitute $x = 0$ back into the original function to find $y$: $y = \ln(0^3 + 3 \cdot 0^2 + 5) = \ln(5) \approx 1.609$

Check for $x = -2$:

1. Substitute $x = -2$ back into the original function to find $y$: $y = \ln((-2)^3 + 3 \cdot (-2)^2 + 5) = \ln(-8 + 12 + 5) = \ln(9) \approx 2.197$

Conclusion

• Relative Minimum: $(0, \ln(5)) \approx (0, 1.609)$
• Relative Maximum: $(-2, \ln(9)) \approx (-2, 2.197)$

Verification with a Graphing Utility

You can verify these values by graphing $y = \ln(x^3 + 3x^2 + 5)$ and observing the points where the graph has relative minima and maxima.

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Do you want more details on the derivative calculation, or have any questions about these steps?

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1. What other methods can we use to find critical points of functions?
2. How does the second derivative test work for determining extrema?
3. Can logarithmic functions have multiple relative extrema?
4. Why do we need to check the domain when dealing with logarithmic functions?
5. How can graphing confirm relative extrema?

Tip: Always check the domain of the function, especially for logarithmic functions, to ensure critical points are within the valid range for $y$.