Question

Draw the graph of a function \(f(x)\) such that \(f^{\prime}(x) \equiv 0,\) and the graph of a function \(g(x)\) such that \(g(3)=0 .\) Summarize in one sentence the essential difference between \(f(x)\) and \(g(x)\) in terms of the concept of stationary point.

Short answer

The essential difference is that \( f(x) \) is stationary everywhere, while \( g(x) \) passes through a single specific stationary point at \( x = 3 \).

Step-by-step solution

Understand the Function f(x)

The function \( f(x) \) has a derivative \( f'(x) \equiv 0 \). This means that the slope of the tangent line to the graph of \( f \) is zero for all \( x \), indicating that \( f(x) \) is a constant function. The graph of \( f(x) \) is a horizontal line.

Graph f(x)

Since \( f(x) \) is a constant function, draw a horizontal line across the graph at any value \( c \) on the \( y \)-axis to represent \( f(x) = c \). There are no specific values required, so any horizontal line will suffice.

Understand the Function g(x)

The function \( g(x) \) should pass through the point \((3, 0)\), meaning \( g(3) = 0 \). This indicates that the curve of the graph intersects the x-axis at \( x = 3 \), possibly a root of \( g(x) \).

Graph g(x)

Ensure the graph of \( g(x) \) passes through the point \( (3, 0) \). There are infinite possible shapes for \( g(x) \) satisfying this condition, but the critical point is that it intersects the x-axis at \( x = 3 \).

Contrast the Functions

The function \( f(x) \) is a constant line because its derivative is always zero, with no change or movement, representing a horizontal stationary point everywhere. In contrast, while \( g(x) \) is not affected everywhere by its slope and might not be constant, it visits a specific point on the x-axis indicating one stationary point visually, where \( x = 3 \).

Key concepts

Derivative

The derivative of a function is a core concept in calculus. It represents the rate at which a function is changing at any given point. In formal terms, the derivative of a function \( f(x) \) with respect to \( x \) is denoted as \( f'(x) \). This notation gives us the slope of the tangent line to the graph of the function at any point \( x \).

When the derivative \( f'(x) \) is equal to zero, it suggests that the function has a horizontal tangent at that point, indicating that the function is not changing.

• If \( f'(x) = 0 \) for all \( x \), the function is constant.
• It shows no growth or decrease, portraying a flat slope eternally.

Understanding derivatives helps us understand the behavior of graphs, including when they rise, fall, or remain constant.

Constant Function

A constant function is one that does not change its value, regardless of the input. The simplest way to identify it is by its form: \( f(x) = c \), where \( c \) is a constant real number. This implies that for every value of \( x \), \( f(x) \) remains the same.

The derivative of a constant function is always zero. This is because there is no change in its value, and thus the slope or rate of change is zero at every point. The graph of a constant function is a horizontal line.

• This horizontal line stretches across the plane, remaining parallel to the x-axis.
• It does not matter what the input value is.

Constant functions are simple yet profound representations in calculus, showing stability and equilibrium.

Stationary Point

A stationary point on a graph is a point where the derivative (slope) of the function is zero. This means there is no change in the function at that particular input. Stationary points can indicate several features:

• Maximum Points: Here, the function reaches its highest value locally.
• Minimum Points: These denote the lowest local value of the function.
• Points of Inflection: At these points, the graph changes its concavity.

For the function \( f(x) \) with a derivative \( f'(x) = 0 \) everywhere, each point on its graph is stationary because the graph is a flat, horizontal line.

In contrast, a function like \( g(x) \) might have isolated stationary points, such as where it crosses the x-axis, making its behavior more complex and varied.

Horizontal Line

A horizontal line is one that runs parallel to the x-axis in a Cartesian coordinate system. This line will intersect the y-axis at a single point, determined by its y-value, such as in a constant function \( f(x) = c \).

Characteristics of a horizontal line include:

• No change in y-value regardless of x.
• Slope of zero, indicating no incline or decline.

Horizontal lines depict situations where a dependent variable remains unaffected by independent variables.

In calculus, when the derivative of a function is zero, it implies the function's graph is a horizontal line, showing consistent and unwavering function values.

X-Intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. This occurs when the dependent variable, usually \( y \), is zero. If a function crosses the x-axis at \( x = a \), then \( f(a) = 0 \).

This point is often a root of the function, indicating that at this specific input, the output value is zero. The function \( g(x) \) with \( g(3) = 0 \) graphically demonstrates that \( x = 3 \) is an x-intercept.

• An x-intercept might signify a turning point or a crossing point where the function value changes sign from positive to negative or vice versa.

X-intercepts are essential for understanding the zeros or roots of functions, which can be critical for broader interpretations of graphs, portraying where functions grow, shrink, or maintain a level plane.