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Let's go step by step to solve the given problem. The function is \( f(x) = a(x - a)(x - 1) \), where \( a \) is a parameter.

### Part (a): Expand the right-hand side and determine \( f'(x) \).

#### Expansion:
We begin by expanding the product \( f(x) = a(x - a)(x - 1) \).

\[
f(x) = a[(x - a)(x - 1)]
\]
First, expand \( (x - a)(x - 1) \):

\[
(x - a)(x - 1) = x^2 - x - ax + a = x^2 - (a + 1)x + a
\]

Thus, the function becomes:

\[
f(x) = a[x^2 - (a + 1)x + a] = a x^2 - a(a + 1)x + a^2
\]

Now, the expanded form of \( f(x) \) is:

\[
f(x) = a x^2 - a(a + 1)x + a^2
\]

#### Derivative \( f'(x) \):
Next, we differentiate \( f(x) = a x^2 - a(a + 1)x + a^2 \) with respect to \( x \).

\[
f'(x) = 2a x - a(a + 1)
\]

So, \( f'(x) = 2a x - a(a + 1) \).

---

### Part (b): Use sign diagrams to determine the sign variation of \( f'(x) \).

We know \( f'(x) = 2a x - a(a + 1) \). To determine the sign of \( f'(x) \), we solve for where \( f'(x) = 0 \).

\[
2a x - a(a + 1) = 0
\]
\[
2a x = a(a + 1)
\]
\[
x = \frac{a(a + 1)}{2a} = \frac{a + 1}{2}
\]

So, \( f'(x) = 0 \) when \( x = \frac{a + 1}{2} \).

#### Sign variation:
- For \( x < \frac{a + 1}{2} \), \( 2a x - a(a + 1) < 0 \), meaning \( f'(x) \) is negative.
- For \( x > \frac{a + 1}{2} \), \( 2a x - a(a + 1) > 0 \), meaning \( f'(x) \) is positive.

Thus, \( f(x) \) is decreasing for \( x < \frac{a + 1}{2} \) and increasing for \( x > \frac{a + 1}{2} \).

---

### Part (c): Use part (b) to examine where \( f(x) \) is increasing or decreasing.

From part (b), we know that:
- \( f(x) \) is decreasing when \( x < \frac{a + 1}{2} \).
- \( f(x) \) is increasing when \( x > \frac{a + 1}{2} \).

Would you like any further clarification or details on any of the steps?

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Here are 5 related questions for further exploration:
1. How does the value of \( a \) affect the point where \( f(x) \) changes from decreasing to increasing?
2. What is the second derivative of \( f(x) \) and what does it say about the concavity of the function?
3. How would you interpret the behavior of the function for negative values of \( a \)?
4. Can you sketch the graph of \( f(x) \) for specific values of \( a \) and observe the turning points?
5. How would you solve for local maxima or minima using the critical points from the derivative?

**Tip**: When analyzing the behavior of functions, always pay attention to the sign of the derivative, as it reveals whether the function is increasing or decreasing.

Given the function f(x) = a(x - a)(x - 1), a is a parameter. a) Expand the right-hand side and determine f'(x). b) Use sign diagrams to determine the sign variation of f'(x). c) Use b. to examine where f(x) is increasing/decreasing.

Explore the function f(x) = a(x - a)(x - 1), where a is a parameter. Learn how to expand the function, find its derivative, and use sign diagrams to determine where it is increasing or decreasing. This problem involves key algebraic and calculus concepts, ideal for students in Grades 11-12.

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