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?

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.