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Chapter 2: Problem 7

Sketch the graph of a function that has the properties described. \(f(2)=1 ; f^{\prime}(2)=0 ;\) concave up for all \(x\)

Short Answer

Expert verified
Plot the point (2,1), draw a horizontal tangent at this point, and sketch a concave up curve.

Step by step solution

01

Identify Key Information

The function has the following key properties: 1. The point \(f(2) = 1\) means it has a value of 1 at x = 2. 2. Its derivative \(f^{\backslash prime}(2) = 0\) means it has a horizontal tangent at x = 2. 3. It is concave up for all x, meaning its second derivative is always positive.
02

Plot Known Point

Start by plotting the point (2, 1) on a coordinate plane. This represents the value of the function at x = 2.
03

Draw Horizontal Tangent

Since the derivative at x = 2 is zero, draw a horizontal tangent line at the point (2, 1). This means the slope of the curve at this point is zero.
04

Sketch Concave Up Curve

Next, sketch a curve that passes through the point (2, 1) and follows a concave up shape. Ensure that the curve is always above the tangent line except at the point (2, 1). A parabola-like shape that opens upwards is a good example.
05

Finalise the Graph

Review the graph to ensure it meets all given criteria: the point (2, 1) is plotted, there is a horizontal tangent at this point, and the curve is concave up for all x.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point Plotting
Point plotting is the first step in sketching any graph. It involves marking specific coordinates on the graph that the function passes through. In our example, we start by plotting the point (2, 1). This tells us that when x is 2, the output of the function, f(x), is 1. Plotting points accurately helps in drawing the overall shape of the graph. Think of it like connecting the dots: each point provides a location that the graph must pass through.
Horizontal Tangent
A horizontal tangent is a line that touches a point on the graph without crossing it. In technical terms, it means the derivative of the function at that point is zero. For our function, the derivative at x = 2 is 0, indicating a horizontal tangent. This translates to a flat section on the graph at (2, 1). Drawing this tangent captures an important aspect of the function's behavior: at x = 2, the slope is zero.
Concavity
Concavity describes how the curve bends. A graph is 'concave up' if it curves like a cup upward, and 'concave down' if it curves like a frown. For the given function, it is concave up for all x. This means the second derivative, which tells us about the concavity, is always positive. When sketching, ensure your curve looks like an upward-facing parabola. This helps visualize the function's increasing rate of change.
Derivative Interpretation
The derivative of a function provides information about the slope of the tangent line at any point on the graph. In our example, knowing the derivative at x = 2 is zero helps identify that the tangent at this point is horizontal. Understanding derivatives allows you to infer critical points and behaviors of the function, such as where it flattens out or changes direction. Integrating these insights ensures an accurate graph representation.

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