To determine the correct answer, let's analyze the function $g(x) = x^4 - 4x^3 + 10$ by finding its critical points and determining the nature of these points.

Step 1: Find the derivative $g'(x)$

The derivative of $g(x)$ is: $g'(x) = 4x^3 - 12x^2$ Factoring out $4x^2$ gives: $g'(x) = 4x^2(x - 3)$ Setting $g'(x) = 0$ to find the critical points: $4x^2(x - 3) = 0$ This results in: $x = 0 \quad \text{or} \quad x = 3$

Step 2: Determine the nature of the critical points

To identify whether each critical point is a local maximum or minimum, we can use the second derivative test.

Find $g''(x)$:

$g''(x) = 12x^2 - 24x$ Factoring gives: $g''(x) = 12x(x - 2)$

Now we evaluate $g''(x)$ at $x = 0$ and $x = 3$:

• At $x = 0$: $g''(0) = 12 \cdot 0 \cdot (0 - 2) = 0$ Since $g''(0) = 0$, the second derivative test is inconclusive for $x = 0$.

• At $x = 3$: $g''(3) = 12 \cdot 3 \cdot (3 - 2) = 12 \cdot 3 \cdot 1 = 36$ Since $g''(3) > 0$, this indicates a local minimum at $x = 3$.

Conclusion

Based on this analysis:

• There is a local minimum at $x = 3$.
• The critical point at $x = 0$ is inconclusive, but it does not indicate a local maximum or minimum for this analysis.

Thus, the correct answer is: The function has only a local minimum at $x = 3$.

Would you like further details on any part of this solution? Here are some related questions for further exploration:

1. How can we confirm if $x = 0$ is a point of inflection?
2. What is the behavior of $g(x)$ as $x \to \infty$ or $x \to -\infty$?
3. How would we approach finding global extrema on a closed interval instead of $(-\infty, \infty)$?
4. Could we use the first derivative test to analyze the nature of critical points instead of the second derivative test?
5. How does the shape of the graph of $g(x)$ help in understanding its extrema?

Tip: For polynomial functions, always start by finding the derivative and setting it to zero to locate critical points. This approach helps identify potential maxima, minima, or points of inflection.