Question

Consider the function \(f(x)=a x^{2}+b x+c,\) with \(a \neq 0 .\) Explain geometrically why \(f\) has exactly one absolute extreme value on \((-\infty, \infty) .\) Find the critical point to determine the value of \(x\) at which \(f\) has an extreme value.

Short answer

Answer: A quadratic function has the form \(f(x) = ax^2 + bx + c\), and its graph is a parabola. Depending on the sign of the leading coefficient \(a\), the parabola either opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)). In both cases, there is only one vertex, which represents the absolute minimum value for upward-opening parabolas and the absolute maximum value for downward-opening parabolas. Therefore, there can only be one absolute extreme value. To find the value of x where it occurs, we can use the vertex formula: \(x_{critical} = -\frac{b}{2a}\).

Step-by-step solution

Recognizing the Shape of Quadratic Functions

The function \(f(x) = ax^2 + bx + c\) is a quadratic function, and its graph is a parabola. There are two cases for the parabolic shape, depending on the sign of the leading coefficient \(a\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), the parabola opens downwards. For both cases, there is only one vertex, which is the point that gives the absolute minimum value for upward-opening parabolas and the absolute maximum value for downward-opening parabolas. This explains why there is only one absolute extreme value for the function over the entire domain.

Finding the Vertex of the Parabola

The vertex of the parabola can be found using the formula \(x_{vertex} = -\frac{b}{2a}\). This represents the \(x\)-coordinate of the vertex (and the critical point) where the extreme value occurs.

Calculating the Critical Point

For the given function \(f(x) = ax^2 + bx + c\), we can calculate the critical point as: \(x_{critical} = -\frac{b}{2a}\) This gives us the value of \(x\) at which the absolute extreme value occurs.

Key concepts

Absolute Extreme Value

The concept of an absolute extreme value refers to the highest or lowest point that a function can attain within a given interval or over its entire domain. In the context of quadratic functions like \(f(x) = ax^2 + bx + c\), with \(a eq 0\), we encounter a special scenario as the function's graph, a parabola, ensures that there is only one such extreme value over its entire domain, which is \(-\infty, \infty\).

For a parabola opening upwards (\(a > 0\)), this absolute extreme is a minimum value, and conversely, for a parabola opening downwards (\(a < 0\)), it's a maximum value. The location of this value can be precisely determined using the vertex of the parabola and calculus to find the critical point, which will be explained in subsequent sections.

Parabolic Shape

When we discuss the parabolic shape particular to quadratic functions, we're talking about the distinctive 'U' shape of their graphs. The direction of the 'U' is determined by the sign of the coefficient \(a\) in the quadratic function \( f(x) = ax^2 + bx + c\).

• If \(a > 0\), the parabola opens upwards, forming what looks like a smile.
• If \(a < 0\), it opens downwards, resembling a frown.

This shape is symmetrical about a vertical line passing through its vertex, which leads us to the concept of the vertex as a point of symmetry and an absolute extreme value on the graph of a quadratic function.

Vertex of a Parabola

The vertex of a parabola is a pivotal concept in understanding quadratic functions. It is the highest or lowest point on the graph of the parabola, depending on whether it opens downwards or upwards, respectively. Geometrically, it's the point where the parabola changes direction.

To find the x-coordinate of the vertex, you can use the formula:
\(x_{vertex} = -\frac{b}{2a}\).
This formula is derived from completing the square or using the differentiation method in calculus. Once you have the x-coordinate, you can substitute it back into the original function to find the y-coordinate, giving you the full coordinates of the vertex. The vertex will represent the location of the absolute extreme value of the function.

Critical Point Calculus

Critical point calculus is a branch of calculus that deals with finding points on a graph where the function's derivative is zero or undefined. These points are critical because they may correspond to local minimums, local maximums, or points of inflection of the function.

In the case of a quadratic function \(f(x) = ax^2 + bx + c\), we utilize its derivative \(f'(x) = 2ax + b\) to locate the critical point. By setting the derivative equal to zero:
\(0 = 2ax + b \Rightarrow x_{critical} = -\frac{b}{2a}\),
we can solve for the value of \(x\). This critical point is also the x-coordinate of the parabola’s vertex, demonstrating its significance in identifying the absolute extreme value of a quadratic function.