Problem 48 - Thomas Calculus

Question

Find the intervals on which the function \(f(x)=a x^{2}+b x+c\) \(a \neq 0,\) is increasing and decreasing. Describe the reasoning behind your answer.

Step-by-Step Solution

Answer

The function increases on \((-\frac{b}{2a}, \infty)\) and decreases on \(( -\infty, -\frac{b}{2a} )\) if \( a > 0 \); opposite for \( a < 0 \).

Step 1: Identify the Critical Point

The function \( f(x) = ax^2 + bx + c \) is a quadratic function. To find intervals of increase and decrease, identify the critical point, which is the vertex of the parabola. The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \).

Step 2: Determine the Direction of the Parabola

Since \( a eq 0 \), the coefficient \( a \) indicates the direction the parabola opens. If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards. This helps determine the increase and decrease intervals relative to the vertex.

Step 3: Analyze Intervals Based on Vertex

For \( a > 0 \), the function is decreasing on the interval \(( -\infty, -\frac{b}{2a} )\) and increasing on \((-\frac{b}{2a}, \infty)\), because the parabola opens upwards. For \( a < 0 \), the function is increasing on \(( -\infty, -\frac{b}{2a} )\) and decreasing on \((-\frac{b}{2a}, \infty)\), due to the parabola opening downwards.

Key Concepts

Critical Points

Critical points are where a function's derivative is zero or undefined. For quadratic functions, to find these points, we focus on the vertex of the parabola. The vertex represents the "peak" or "valley" of the parabola. It's a special point because that's where the function changes from increasing to decreasing, or vice versa.

For a quadratic function given by \(f(x) = ax^2 + bx + c\), the x-coordinate of the vertex, which is the critical point, can be found using the formula:

• \(x = -\frac{b}{2a}\)

This formula is derived from setting the derivative of the quadratic to zero and solving for \(x\). This is because the derivative \( f'(x) = 2ax + b\) represents the slope of the tangent to the parabola, and setting \( f'(x) = 0 \) finds where this slope is flat, which occurs at the vertex.

Understanding the location of the critical point is crucial for analyzing how the function behaves to the left and right of this point.

Intervals of Increase and Decrease

In calculus, determining where a function increases or decreases is essential for understanding its overall behavior. By knowing the intervals of increase and decrease, you can graph the function more accurately and gain insight into its properties.

For our quadratic function \(f(x) = ax^2 + bx + c\), the behavior depends on the coefficient \(a\):

• If \(a > 0\): The parabola opens upwards. The function decreases on the interval \((-\infty, -\frac{b}{2a})\) and increases on \((-\frac{b}{2a}, \infty)\).
• If \(a < 0\): The parabola opens downwards. The function increases on the interval \((-\infty, -\frac{b}{2a})\) and decreases on \((-\frac{b}{2a}, \infty)\).

Each of these intervals is relative to the vertex. The examination of these intervals is crucial for sketching the parabola, because it tells you where the function reaches its lowest or highest point (the vertex), and how it behaves on either side.

Parabola Vertex

Understanding the vertex of a parabola is key when examining quadratic functions. The vertex is the "turning point" of the parabola.For a quadratic function of the form \(f(x) = ax^2 + bx + c\):

• The vertex is given as \(\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\).

To find the y-coordinate of the vertex, substitute the x-value \(-\frac{b}{2a}\) back into the function to get the point \(f\left(-\frac{b}{2a}\right)\). This will provide a full understanding of where the vertex is located on the graph.

The vertex tells us whether the function has a minimum or maximum value:

• Minimum: Occurs if the parabola opens upwards, so \(a > 0\).
• Maximum: Occurs if the parabola opens downwards, so \(a < 0\).

The vertex acts as a guide for determining the intervals of increase and decrease, as it divides the parabola into two symmetrical areas. Knowing this helps us not only graph the function precisely but also understand its behavior deeply.