Question

In Exercises, sketch a graph of a function \(f\) having the given characteristics. \begin{aligned} &f(2)=f(4)=0 \\ &f^{\prime}(x)<0 \text { if } x<3 \\ &f^{\prime}(3)=0 \\ &f^{\prime}(x)>0 \text { if } x>3 \\ &f^{\prime}(x)>0 \end{aligned}

Short answer

A rough sketch of the function using these characteristics would start from the top left (decreasing), cross the x-axis at \(x=2\), reach a local minimum at \(x=3\), cross the x-axis again at \(x=4\), and then continue upward (increasing) to the top right.

Step-by-step solution

Identify Roots

From \(f(2)=f(4)=0\), we see that our function has roots at \(x=2\) and \(x=4\). So, we mark these points on our x-axis.

Understand the Slope: Part I

From \(f^{\prime}(x)<0\) for \(x<3\), we know that the derivative (or the slope of the tangent line to the function) of our function is negative for values of \(x\) less than 3. This means that the function is decreasing on the interval \((-\infty, 3)\).

Identify Turning Point

Now, from \(f^{\prime}(3)=0\), we see that the slope of the tangent at x=3 is zero. We know this point is called a stationary point. Since the function changes from decreasing before x=3 to increasing after x=3, this stationary point is a local minimum.

Understand the Slope: Part II

Finally, from \(f^{\prime}(x)>0\) for \(x>3\), we know that our function is increasing on the interval \((3, +\infty)\). This implies that after the local minimum at \(x=3\), the function is increasing and hence has positive slope.

Key concepts

Graph of a function

The graph of a function is a visual representation of all the points that satisfy the function. Imagine plotting a set of points on a coordinate plane, where each point corresponds to an input value and its corresponding output value. This is essentially what the graph does. The x-axis typically represents the input values, while the y-axis represents the function's output. Drawing a graph helps us understand the behavior of the function, such as where it increases, decreases, or remains constant.

When sketching a function, noting specific features like the roots, stationary points, and changes in slope enhances accuracy. These characteristics provide insight into how the graph should be structured, allowing us to visualize its shape and curves.

Derivative

The derivative of a function, denoted as \(f'(x)\), is a fundamental aspect in calculus that signifies the rate of change of the function. In simple terms, it tells us how the function is behaving at any particular point, by providing the slope of the tangent line at that point.

There are several important interpretations of the derivative:

• A positive derivative \(f'(x) > 0\) indicates that the function is increasing at that point.
• A negative derivative \(f'(x) < 0\) suggests that the function is decreasing.
• A derivative of zero \(f'(x) = 0\) means the function has a stationary point, which could be a local maximum, minimum, or a point of inflection.

Understanding the derivative is crucial as it provides deeper insight into the function's graph, specifically how the curve moves and changes.

Roots

Roots of a function are the x-values where the function equals zero, meaning \(f(x) = 0\). In the context of a graph, roots are where the curve intersects the x-axis. These points are essential in sketching a graph since they serve as anchor points and give insight into the function's behavior near these areas.

To find the roots, set the function equation equal to zero and solve for \(x\). For example, if given \(f(2) = 0\) and \(f(4) = 0\), the roots are at \(x = 2\) and \(x = 4\). These points are critical because any change in the sign of the function (from positive to negative, or vice versa) often occurs at or near these root locations. Knowing the roots helps in determining the function's intervals of increase, decrease, and other behaviors.

Stationary point

A stationary point occurs where the derivative of a function \(f'(x)\) is zero, indicating that the slope of the tangent at that point is horizontal. At a stationary point, the function does not immediately increase or decrease. Instead, it can be a local minimum, a local maximum, or a point of inflection.

In the example, where \(f'(3) = 0\), \(x = 3\) is a stationary point. Since the derivative changes from negative to positive, this stationary point represents a local minimum. This means that at \(x = 3\), the function dips to its lowest in the nearby vicinity before rising again.

Identifying stationary points is crucial because they help in understanding the overall shape of a function's graph and determining whether the graph will rise or fall after the point.