Problem 56 - Calculus of a Single Variable

Question

Think About It In Exercises \(53-56\) , sketch the graph of a function \(f\) having the given characteristics. $$ \begin{array}{l}{f(0)=f(2)=0} \\ {f^{\prime}(x)<0 \text { for } x<1} \\\ {f^{\prime}(1)=0} \\ {f^{\prime}(x)>0 \text { for } x>1} \\ {f^{\prime \prime}(x)>0}\end{array} $$

Step-by-Step Solution

Answer

Given the characteristics of the function \(f\), it starts at the point (0,0), decreases until \(x=1\), at which point it has a horizontal tangent, and then begins to increase till (2,0). The function is concave up throughout its domain.

Step 1: Plotting the x-intercepts

Since \(f(0)=f(2)=0\), we know that the graph of this function passes through the points (0,0) and (2,0). Plot these points on the graph.

Step 2: Sketching the behavior of the first derivative

When \(f^{\prime}(x)<0\) for \(x<1\), we know that the function is decreasing. This means that the slope of the function is negative when \(x<1\). Sketch the graph decreasing from the point (0,0) till \(x=1\). At \(x=1\), \(f^{\prime}(1)=0\) which means there is a horizontal tangent at \(x=1\). Lastly, since \(f^{\prime}(x)>0\) for \(x>1\), the slope of the function is positive for \(x>1\). Sketch the graph as increasing after \(x=1\) till the point (2,0).

Step 3: Sketching the behavior of the second derivative

Finally consider the condition that \(f^{\prime\prime}(x)>0\), which means the function is concave up everywhere. Make sure that the curve of this function is drawn as concave up throughout the domain.

Key Concepts

x-intercepts

Understanding x-intercepts is fundamental in graphing. X-intercepts are where the graph of a function crosses the x-axis. These points indicate the values of \(x\) for which \(f(x) = 0\). In our example, we are given that \(f(0) = 0\) and \(f(2) = 0\), which means the x-intercepts of the function are at \((0,0)\) and \((2,0)\). Identifying these points is crucial, as they act as anchors for sketching the rest of the graph. The graph must pass through both these intercepts, giving the sketch its specific bounds on the x-axis!

First Derivative Test

The First Derivative Test helps us understand the increasing and decreasing behavior of a function. It involves analyzing the first derivative \(f'(x)\) of the function, which gives us the slope. Here's how it works:

• If \(f'(x) < 0\), the function is decreasing.
• If \(f'(x) > 0\), the function is increasing.
• If \(f'(x) = 0\), there might be a horizontal tangent, which can indicate a local minimum or maximum.

In our problem, \(f'(x) < 0\) for \(x < 1\), which tells us the function is decreasing up to \(x = 1\). At \(x = 1\), \(f'(1) = 0\), indicating a flat spot or a potential turning point. After this, \(f'(x) > 0\) for \(x > 1\), suggesting that the graph turns upward from this point. This analysis helps us sketch the function’s general shape.

Concavity

Concavity in a function is illustrated through the second derivative \(f''(x)\). It indicates how the curvature of the graph is oriented:

• If \(f''(x) > 0\), the function is "concave up" like a cup.
• If \(f''(x) < 0\), the function is "concave down" like a frown.

In our scenario, \(f''(x) > 0\) for all x, telling us that the graph is always concave up. This means the curve opens upwards throughout the entire graph. The consistent concavity impacts how we perceive changes in slope, as it ensures there's a smooth transition, making the graph resemble a gentle 'U' shape.

Slope Analysis

Slope Analysis provides great insight into the direction and steepness of a function. It combines both the first derivative and the nature of the curve to guide the sketching process.Here's what we discerned from the problem:

• Initially, for \(x < 1\), the slope is negative, indicating a decline up to \(x = 1\).
• At \(x = 1\), the slope turns zero, suggesting a turning point or the presence of a tangent flatline.
• For \(x > 1\), the positive slope reflects an increase.
• Since \(f''(x) > 0\), the graph is bending upwards continuously, reinforcing the upward slope as a steady rise post-\(x = 1\).

Understanding the slope at different sections of the domain is critical for constructing an accurate visual representation of the function.