Problem 16 - Calculus: Early Transcendentals

Question

Let \(f(x)=a x^{2}+b x+c\) with \(a \neq 0\). Show that \(f\) has exactly one critical point using the first derivative test. Give conditions on a and b which guarantee that the critical point will be a maximum. It is possible to see this without using calculus at all; explain.

Step-by-Step Solution

Answer

The function has one critical point at \(x = -\frac{b}{2a}\). It is a maximum if \(a < 0\).

Step 1: Find the First Derivative

To find the critical points of the function, we need to determine where the first derivative equals zero. Let's differentiate the function: \[ f(x) = ax^2 + bx + c \]The first derivative, \(f'(x)\), is given by:\[ f'(x) = \frac{d}{dx}(ax^2 + bx + c) = 2ax + b \]

Step 2: Solve for Critical Points

To find the critical points, set the first derivative equal to zero:\[ 2ax + b = 0 \]Solving for \(x\), we find:\[ x = -\frac{b}{2a} \]This shows the function has exactly one critical point at \(x = -\frac{b}{2a}\).

Step 3: Perform the First Derivative Test

To determine the nature of the critical point, evaluate the sign of the first derivative on either side of \(x = -\frac{b}{2a}\):- If \(x < -\frac{b}{2a}\), then \(f'(x)\) has the sign of \(2a\), meaning it depends on whether \(a > 0\) or \(a < 0\).- If \(x > -\frac{b}{2a}\), then \(f'(x)\) again has the sign of \(2a\).If \(a > 0\), the function decreases before and increases after, making it a minimum. If \(a < 0\), the function increases before and decreases after, making it a maximum.

Step 4: Non-Calculus Explanation

The function \(f(x) = ax^2 + bx + c\) is a parabola. If \(a > 0\), the parabola opens upwards and its vertex is the minimum point. If \(a < 0\), the parabola opens downwards and its vertex is the maximum point. Thus, without calculus, we see \(a < 0\) is required for the critical point to be a maximum.

Key Concepts

First Derivative Test

To understand how the First Derivative Test helps in analyzing critical points, it's important to first grasp the idea of the derivative itself.
The derivative of a function expresses the slope or the rate of change of the function at a given point.Here's how the First Derivative Test is applied:

• Solve for the derivative of the function, as we did with the quadratic function to get \(f'(x) = 2ax + b\).
• Set the derivative equal to zero to find critical points. Solving \(2ax + b = 0\) leads us to the critical point \(x = -\frac{b}{2a}\).
• Examine the derivatives on either side of our critical point. This will tell us whether the function is increasing or decreasing on that interval.

In simple terms, if the derivative changes from positive to negative at the critical point, the function has a local maximum. Conversely, if it changes from negative to positive, you have a local minimum. This change in sign is a key indicator of the nature of the critical point.

Critical Points

Critical points are extremely important in understanding the behavior of functions. In our example with the quadratic function \(f(x) = ax^2 + bx + c\), we saw that there is exactly one critical point, which we found as \(x = -\frac{b}{2a}\).
To find critical points:

• Calculate the first derivative as before.
• Set the first derivative equal to zero and solve for \(x\).

For quadratic functions like ours, the critical point corresponds to the vertex of the parabola represented by the function. This means it is where the highest or lowest point of the curve occurs, depending on the parabola's orientation.
Critical points allow us to determine where the function reaches these extreme points, which is crucial for solving optimization problems or simply understanding the function's graph.

Parabola Properties

The quadratic function discussed forms a parabola when graphed.
Understanding the properties of a parabola helps in visualizing the solution and comprehension of the First Derivative Test and critical points.Key properties of parabolas include:

• The direction it opens depends on the coefficient \(a\).
• If \(a > 0\), the parabola opens upwards, resembling a smiling face.
• If \(a < 0\), it opens downwards, like a frowning face.
• The vertex of the parabola is the minimum or maximum point, depending on its orientation.

For the quadratic function \(f(x) = ax^2 + bx + c\), knowing whether the parabola opens upwards or downwards assists us in determining if the critical point found is a minimum or a maximum.
Thus, even without calculus, observing that \(a < 0\) indicates a downward opening parabola gives us the maximum at the vertex, which aligns with the derivative's findings.