Question

Graph a function \(f(x)\) that gives rise to the derivative function \(f^{\prime}(x)=0 .\) Then graph a function \(g(x)\) characterized by \(g^{\prime}\left(x_{0}\right)=0\)

Short answer

Graph \( f(x) = c \) for a constant line; graph \( g(x) = (x-x_0)^2+c \) for a parabola with horizontal tangent at \( x_0 \).

Step-by-step solution

Understanding the Derivative

The derivative of a function at any point gives us the slope of the tangent line at that point. If a function has a derivative of zero everywhere, it means its slope is zero everywhere, implying that the function itself is a horizontal line.

Graphing the Constant Function

Since the derivative of a constant function is zero everywhere, choose a constant function like \( f(x) = c \), where \( c \) is any real number. This line is horizontal with a slope of 0 at all points.

Graphing the Function

On a coordinate plane, draw a horizontal line to represent \( f(x) = c \). This line will have the property that \( f^{\prime}(x) = 0 \) for all \( x \).

Understanding Critical Points

A function with a derivative of zero at a specific point \( x_0 \) has a critical point, which could be a local maximum, minimum, or a point of inflection depending on the behavior of the function around \( x_0 \).

Graphing a Parabolic Function

An example of such a function is \( g(x) = (x-x_0)^2+c \), where \( x_0 \) is a constant point at which the derivative is zero, and \( c \) is any real number. At \( x_0 \), the slope will be zero, indicating a horizontal tangent line.

Graphing the Function with Critical Point

Plot the function \( g(x) = (x-x_0)^2+c \) on the coordinate plane. It is a parabola opening upwards, and at \( x = x_0 \), the tangent line is horizontal because the derivative is zero.

Key concepts

Critical Points in Calculus

In calculus, a critical point of a function occurs where its derivative is zero or undefined. Critical points are significant because they are potential locations for local maxima, minima, or points of inflection. These points can be identified by finding where the derivative of a function equals zero or does not exist.

The importance of critical points includes:

• Identifying where a function changes direction, helping to sketch graphs.
• Determining the behavior of the function around these points to classify them as a maximum, minimum, or inflection point.
• Assisting in optimization problems where finding the highest or lowest values is essential.

For instance, consider the function \( g(x) = (x-x_0)^2 + c \). Here, the derivative \( g'(x) \) is zero at \( x = x_0 \), signifying a critical point. This point is instrumental in understanding the behavior of the function since, at \( x_0 \), the parabola has a horizontal tangent indicating a local minimum.

Graphing Functions

Graphing functions is a crucial technique for visualizing mathematical expressions and their properties. It allows us to see where functions increase, decrease, or remain constant, providing an intuitive understanding of their behavior.

When graphing a function like \( f(x) = c \), which is a constant function, we see a straight horizontal line. This represents that no matter the input \( x \), the output remains the same. This type of visual representation is simple yet powerful.

For functions with critical points, such as \( g(x) = (x-x_0)^2 + c \), the graph will show a parabola. Graphing provides insights such as:

• The shape of the curve and how it opens.
• The location of critical points where the slope is zero.
• The consistent slope of zero along certain paths, like horizontal lines for constant functions.

Understanding these graphical properties is essential for solving equations and predicting behavior across different contexts.

Constant Functions

Constant functions are some of the simplest types of functions in mathematics. A constant function is defined as \( f(x) = c \), where \( c \) is a constant value, and \( f(x) \) remains unchanged regardless of the value of \( x \).

One of the key properties of a constant function is that its derivative is zero everywhere because the slope of a constant line is flat. Therefore, if a function's derivative \( f'(x) = 0 \) at every point, it implies the function is constant.

Studying constant functions helps in understanding more complex behaviors in calculus by:

• Serving as a basis for comparison to non-constant functions.
• Representing stationary processes without change over time or space.
• Providing foundational knowledge for understanding undifferentiated states of functions.

Overall, constant functions illustrate the idea of constancy and stability in mathematical terms, where the output is unaffected by changes in the input, mirrored by their horizontal graphing on a coordinate plane.