Problem 1 - Calculus Single Variable

Question

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=(x-2)^{2}+3 $$

Step-by-Step Solution

Answer

The point \((2, 3)\) is a local minimum for \(f(x)\).

Step 1: Understand the function form

The given function is defined as \( f(x) = (x-2)^2 + 3 \). This is a quadratic function, which represents a parabola opening upwards because the coefficient of \( (x-2)^2 \) is positive.

Step 2: Identify the vertex

For the quadratic function \( f(x) = a(x-h)^2 + k \), the vertex is located at the point \((h, k)\). In this function, \( a = 1 \), \( h = 2 \), and \( k = 3 \). Therefore, the vertex is at the point \((2, 3)\).

Step 3: Determine local extrema points

Since the parabola opens upwards, the vertex at \((2, 3)\) is the lowest point. Hence, this point represents the local minimum of the function \( f(x) \). There are no local maxima for this parabola since it opens upwards.

Key Concepts

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function where the highest degree of the variable is 2. They are generally written in the standard form as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the context of our exercise, the quadratic function given is \( f(x) = (x-2)^2 + 3 \). Here, the function is expressed in a slightly different form: \( a(x-h)^2 + k \), known as the vertex form. Each part of this function tells us important information:

• \( a \): describes the direction of the parabola. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
• \((h, k)\): gives the coordinates of the vertex of the parabola.

This function simplifies our work in identifying the parabola's crucial attributes, like its vertex and direction.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on the orientation of the parabola. For the function \( f(x) = (x-2)^2 + 3 \), the vertex can be easily identified thanks to its vertex form, \( a(x-h)^2 + k \).

• Here, \( h = 2 \) and \( k = 3 \).
• The vertex is the point \((h, k)\), so it's \((2, 3)\) for our function.

This vertex at \((2, 3)\) represents a single point that signifies not only the turn of the parabola but also the minimum value the function can achieve in the context of an upward-opening parabola.

What is a Local Minimum?

A local minimum is the lowest point in a particular section of a graph of a function, meaning there are no values less than it in its immediate vicinity. In our exercise, we explored \( f(x) = (x-2)^2 + 3 \). The graph of this function is a parabola that opens upwards. This tells us:

• Since the parabola opens upwards, the vertex at \((2, 3)\) is the lowest point.
• This point, \((2, 3)\), is where the local minimum occurs.

There is no local maximum in this case, because an upward-opening parabola only dips down to its lowest point before rising indefinitely on either side. Understanding the concept of local minimum helps in identifying key points on graphs and optimizing solutions in real-world problems.