To find where the local maximum of the function $g(x) = x^4 - 6x^2 + 9$ occurs, we need to follow these steps:

1. Find the first derivative of the function:

$g'(x) = \frac{d}{dx}(x^4 - 6x^2 + 9)$ $g'(x) = 4x^3 - 12x$

2. Set the first derivative equal to zero to find critical points:

$4x^3 - 12x = 0$ Factor the equation: $4x(x^2 - 3) = 0$ Set each factor equal to zero: $4x = 0 \quad \Rightarrow \quad x = 0$ $x^2 - 3 = 0 \quad \Rightarrow \quad x = \pm\sqrt{3}$

So, the critical points are $x = 0$, $x = \sqrt{3}$, and $x = -\sqrt{3}$.

3. Determine the nature of each critical point using the second derivative:

Find the second derivative $g''(x)$: $g''(x) = \frac{d}{dx}(4x^3 - 12x)$ $g''(x) = 12x^2 - 12$

Evaluate $g''(x)$ at each critical point:

• At $x = 0$: $g''(0) = 12(0)^2 - 12 = -12 \quad (\text{Negative, indicating a local maximum})$
• At $x = \sqrt{3}$ and $x = -\sqrt{3}$: $g''(\sqrt{3}) = 12(\sqrt{3})^2 - 12 = 12(3) - 12 = 36 - 12 = 24 \quad (\text{Positive, indicating local minima})$

4. Conclusion:

The local maximum occurs at $x = 0$.

Would you like to explore more details, or do you have any questions?

Here are some related questions:

1. How do you determine if a critical point is a local maximum, minimum, or a saddle point?
2. What is the significance of the second derivative test in finding local extrema?
3. Can a function have more than one local maximum or minimum?
4. How do the coefficients of a polynomial function influence the location and nature of critical points?
5. What is the relationship between the degree of a polynomial and the number of critical points?

Tip: The second derivative test is a quick way to determine the nature of a critical point: if $g''(x)$ is negative at a critical point, it's a local maximum; if positive, it's a local minimum.