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The given function is:

\[
p(x) = -x^3 + 6x^2 - 9x + 12
\]

We are asked to find the value of \(x\) at which the function reaches a relative maximum. To do this, we need to find the critical points of the function by following these steps:

### 1. Take the derivative of the function.
The first derivative of \( p(x) \), denoted as \( p'(x) \), will help us find the critical points:

\[
p'(x) = \frac{d}{dx} \left( -x^3 + 6x^2 - 9x + 12 \right)
\]

\[
p'(x) = -3x^2 + 12x - 9
\]

### 2. Set the derivative equal to zero to find the critical points.
We solve \( p'(x) = 0 \):

\[
-3x^2 + 12x - 9 = 0
\]

To simplify, divide through by \(-3\):

\[
x^2 - 4x + 3 = 0
\]

Factor the quadratic equation:

\[
(x - 3)(x - 1) = 0
\]

So, \( x = 3 \) and \( x = 1 \).

### 3. Determine which point is a maximum.
To confirm which critical point is a maximum, we take the second derivative of \( p(x) \):

\[
p''(x) = \frac{d}{dx} (-3x^2 + 12x - 9)
\]

\[
p''(x) = -6x + 12
\]

Now, evaluate the second derivative at \( x = 1 \) and \( x = 3 \):

- For \( x = 1 \):

\[
p''(1) = -6(1) + 12 = 6
\]

Since \( p''(1) > 0 \), \( x = 1 \) is a minimum point.

- For \( x = 3 \):

\[
p''(3) = -6(3) + 12 = -6
\]

Since \( p''(3) < 0 \), \( x = 3 \) is a maximum point.

### Conclusion:
The function reaches a relative maximum at \( x = 3 \).

---

Would you like more details or have any other questions?

Here are some related questions to deepen your understanding:
1. How do we find critical points for any polynomial function?
2. What is the significance of the second derivative in determining concavity?
3. Can a function have more than one relative maximum or minimum?
4. How does changing the coefficients in a cubic function affect its graph?
5. Why is dividing the equation by \(-3\) important when solving the derivative?

**Tip:** Always check both the first and second derivatives to correctly classify critical points as maxima, minima, or points of inflection.

Fungsi p(x) = -x^3 + 6x^2 - 9x + 12 akan mencapai balik maksimum pada saat x = ....

Learn how to find the maximum point of the cubic function -x^3 + 6x^2 - 9x + 12 using derivatives. This problem involves calculating the first and second derivatives to determine the relative maximum. Suitable for Grades 11-12.

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