The function in question is $f(x) = ax^2 + bx + c$, where $a \neq 0$. To determine where the function is increasing and decreasing, we follow these steps:

Step 1: Find the first derivative

The first derivative of $f(x)$ is: $f'(x) = 2ax + b$ This tells us the slope of the tangent line to the curve at any point $x$.

Step 2: Set the derivative equal to zero to find critical points

To find the critical point where the function changes from increasing to decreasing (or vice versa), set $f'(x) = 0$: $2ax + b = 0$ Solving for $x$: $x = -\frac{b}{2a}$ This is the $x$-coordinate of the vertex of the parabola, and it is where the function's behavior changes.

Step 3: Determine intervals of increasing and decreasing

The sign of $f'(x)$ (the derivative) determines whether the function is increasing or decreasing:

• If $f'(x) > 0$, the function is increasing.
• If $f'(x) < 0$, the function is decreasing.

Case 1: $a > 0$ (upward-opening parabola)

• The function will decrease to the left of $x = -\frac{b}{2a}$ because $f'(x) < 0$ when $x < -\frac{b}{2a}$.
• The function will increase to the right of $x = -\frac{b}{2a}$ because $f'(x) > 0$ when $x > -\frac{b}{2a}$.

Thus, when $a > 0$:

• The function is decreasing on $(-\infty, -\frac{b}{2a})$.
• The function is increasing on $(-\frac{b}{2a}, \infty)$.

Case 2: $a < 0$ (downward-opening parabola)

• The function will increase to the left of $x = -\frac{b}{2a}$ because $f'(x) > 0$ when $x < -\frac{b}{2a}$.
• The function will decrease to the right of $x = -\frac{b}{2a}$ because $f'(x) < 0$ when $x > -\frac{b}{2a}$.

Thus, when $a < 0$:

• The function is increasing on $(-\infty, -\frac{b}{2a})$.
• The function is decreasing on $(-\frac{b}{2a}, \infty)$.

Part 1: Answering the Multiple-Choice Question

When $a > 0$, $f$ is increasing on:

$\boxed{\left( -\frac{b}{2a}, \infty \right)}$ because $f'(x)$ is positive in this interval.

$f$ is decreasing on:

$\boxed{\left( -\infty, -\frac{b}{2a} \right)}$ because $f'(x)$ is negative in this interval.

Part 2: When $a < 0$, $f$ is increasing on:

$\boxed{\left( -\infty, -\frac{b}{2a} \right)}$ because $f'(x)$ is positive in this interval.

$f$ is decreasing on:

$\boxed{\left( -\frac{b}{2a}, \infty \right)}$ because $f'(x)$ is negative in this interval.

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Would you like more details or have any further questions?
Here are 5 related questions to expand your understanding:

1. How does the second derivative of a quadratic function affect concavity?
2. How can the vertex form of a quadratic function help determine the intervals of increase and decrease?
3. What is the significance of the discriminant in a quadratic function?
4. How can we apply the first derivative test to determine local extrema?
5. How does completing the square help in finding the vertex of a quadratic function?

Tip: The first derivative tells you the slope of the tangent line, which directly indicates whether the function is increasing or decreasing.