To find the x-coordinates of all local maxima using the second derivative test, follow these steps:

1. Given Function:

The function is: $f(x) = x^9 - 3x^7 + 2$

2. Find the first derivative $f'(x)$:

$f'(x) = \frac{d}{dx}(x^9 - 3x^7 + 2) = 9x^8 - 21x^6$

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

$9x^8 - 21x^6 = 0$ Factor the equation: $x^6(9x^2 - 21) = 0$ $x^6(3x^2 - 7) = 0$ This gives two possibilities:

• $x^6 = 0$ ⟹ $x = 0$
• $3x^2 - 7 = 0$ ⟹ $x^2 = \frac{7}{3}$ ⟹ $x = \pm \sqrt{\frac{7}{3}}$

Thus, the critical points are: $x = 0, \quad x = \pm \sqrt{\frac{7}{3}}$

4. Find the second derivative $f''(x)$:

$f''(x) = \frac{d}{dx}(9x^8 - 21x^6) = 72x^7 - 126x^5$

5. Apply the second derivative test:

• At $x = 0$: $f''(0) = 72(0)^7 - 126(0)^5 = 0$ Since the second derivative is zero, the test is inconclusive at $x = 0$. We cannot determine if this is a local maximum using the second derivative test alone.

• At $x = \sqrt{\frac{7}{3}}$: $f''\left(\sqrt{\frac{7}{3}}\right) = 72\left(\sqrt{\frac{7}{3}}\right)^7 - 126\left(\sqrt{\frac{7}{3}}\right)^5$ Without simplifying, observe that the degree of the powers makes the second derivative negative (since both terms have odd powers and positive coefficients for this positive $x$), so: $f''\left(\sqrt{\frac{7}{3}}\right) < 0$ Thus, $x = \sqrt{\frac{7}{3}}$ is a local maximum.

• At $x = -\sqrt{\frac{7}{3}}$: Similarly, we check: $f''\left(-\sqrt{\frac{7}{3}}\right) = 72\left(-\sqrt{\frac{7}{3}}\right)^7 - 126\left(-\sqrt{\frac{7}{3}}\right)^5$ The second derivative will also be negative here (due to the same reasoning as above), so: $f''\left(-\sqrt{\frac{7}{3}}\right) < 0$ Thus, $x = -\sqrt{\frac{7}{3}}$ is also a local maximum.

Conclusion:

The x-coordinates of the local maxima are: $x = \pm \sqrt{\frac{7}{3}}$

Would you like more details or further explanations? Here are 5 related questions to expand on this concept:

1. How do you simplify $f''(x)$ to evaluate it at specific points?
2. Why is the second derivative test inconclusive when $f''(x) = 0$?
3. How do we determine whether a critical point is a maximum or a minimum using concavity?
4. Can we use a graph to verify the location of maxima and minima for this function?
5. What happens if we use the first derivative test instead of the second derivative test?

Tip: When the second derivative test is inconclusive, try using the first derivative test or higher-order derivatives.