Question

Concavity of parabolas Consider the general parabola described by the function \(f(x)=a x^{2}+b x+c .\) For what values of \(a, b\) and \(c\) is \(f\) concave up? For what values of \(a, b,\) and \(c\) is \(f\) concave down?

Short answer

Answer: The general parabola f(x) = ax^2 + bx + c is concave up when a > 0, and concave down when a < 0. The coefficients b and c do not influence the concavity.

Step-by-step solution

Find the first and second derivatives of f(x)

To find out the concavity of the parabola, we need to find the second derivative of the function \(f(x)\) and check its sign. First, let's find the first and second derivatives of \(f(x)\). Given function is \(f(x) = ax^2 + bx + c\). Using the power rule to find the first derivative: \(f'(x) = \frac{d}{dx}(ax^2 + bx + c) = 2ax + b\). Now, finding the second derivative by applying the power rule to \(f'(x)\): \(f''(x) = \frac{d}{dx}(2ax + b) = 2a.\)

Determine the concavity using the second derivative

For a parabola to be concave up, the second derivative must be positive (i.e., \(f''(x) > 0\)), and for a parabola to be concave down, the second derivative must be negative (i.e., \(f''(x) < 0\)). Since our second derivative is constant, we can use it directly to determine our conditions. For concave up: \(f''(x) = 2a > 0 \Rightarrow a > 0.\) For concave down: \(f''(x) = 2a < 0 \Rightarrow a < 0.\)

Conclusion

Thus, for the general parabola \(f(x) = ax^2 + bx + c\) to be concave up, the coefficient \(a\) must be greater than zero, i.e., \(a > 0\). And for the parabola to be concave down, the coefficient \(a\) must be less than zero, i.e., \(a < 0\). The coefficients \(b\) and \(c\) do not influence the concavity.

Key concepts

Second Derivative Test

The second derivative test is a critical tool for determining the concavity of a function. Here's what it entails:
In calculus, we often want to know where a function is curving upwards or downwards. This insight helps us understand the behavior of the function beyond just its increasing or decreasing nature. By taking the second derivative of the function, denoted as \( f''(x) \), we obtain a measure of how the function's slope is changing.

• If \( f''(x) > 0 \), the function is concave up, which visually means the graph forms a basin that could 'hold water'.
• If \( f''(x) < 0 \), the function is concave down, resembling an upside-down basin or arch.

In the case of a parabola described by \( f(x) = ax^2 + bx + c \), as in the exercise, the second derivative simplifies to \( 2a \). Since \( a \) is a constant, the concavity of the parabola is solely dependent on the sign of \( a \). If \( a > 0 \), the parabola opens upwards and is concave up. Conversely, if \( a < 0 \), it opens downwards and is concave down.

Parabola Concavity

Understanding parabola concavity is essential in graph interpretation and calculus problems. Concavity refers to the direction in which a parabola opens. It's a fundamental property defined by the sign of the quadratic term's coefficient in a parabola's equation.
For the standard parabola equation \( f(x) = ax^2 + bx + c \), the concavity depends entirely on the coefficient \( a \):

• When \( a > 0 \), the parabola opens upwards—this is what we mean by concave up. Think of it as a smiling face.
• When \( a < 0 \), the parabola opens downwards, which is referred to as concave down, akin to a frown.

The coefficients \( b \) and \( c \) in the parabola equation affect the location and height of the vertex but don't influence its concavity. This peculiarity makes the coefficient \( a \) the sole determiner of a parabola's curve direction.

First Derivative

The first derivative of a function, denoted as \( f'(x) \), is a foundational concept in calculus that indicates the slope or rate of change of the function at any given point. In simpler terms, it tells us how fast the function's value is increasing or decreasing as you move along the x-axis.
In our parabola \( f(x) = ax^2 + bx + c \), calculating the first derivative by applying the power rule gives us \( f'(x) = 2ax + b \). This derivative is valuable for several reasons:

• It helps locate the critical points of the function, where \( f'(x) = 0 \), indicating potential maximum, minimum, or inflection points.
• It reveals the intervals on which the function is increasing \( f'(x) > 0 \) or decreasing \( f'(x) < 0 \).

However, it doesn't directly tell us about the concavity; this is where the second derivative comes into play, as previously discussed.

Coefficient Analysis

Coefficient analysis is a method of examining the coefficients in the equation of a function to determine its key properties. In our example with the parabola \( f(x) = ax^2 + bx + c \), this simple analysis yields profound insights into the graph's shape through the leading coefficient \( a \).
Here's how this plays out:

• If \( a > 0 \), the parabola's arms extend upward, and we have a minimum point at the vertex of the parabola. In this case, the parabola is concave up which can also be verified by the second derivative test.
• If \( a < 0 \), the arms point downward, and the parabola has a maximum point at the vertex. Thus, the graph is concave down.

Coefficients \( b \) and \( c \) can shift the parabola left, right, up, or down, but they don't affect the essential 'U' or 'n' shape dictated by \( a \). Coefficient analysis often complements other calculus techniques to provide a complete picture of the graph's behavior.