$$  $$$$ x^9-3x^7+2 $$Find the X-cordinates of all local maxima using the second derivative test

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.

Find the X-cordinates of all local maxima using the second derivative test for the function x^9 - 3x^7 + 2.

Learn how to find the x-coordinates of all local maxima for the function x^9 - 3x^7 + 2 using the second derivative test. This solution involves calculating first and second derivatives, identifying critical points, and applying the second derivative test. Suitable for Grade 12 or college calculus students.

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