Problem 55 - Calculus of a Single Variable

Question

Think About It In Exercises \(53-56\) , sketch the graph of a function \(f\) having the given characteristics. $$ \begin{array}{l}{f(2)=f(4)=0} \\ {f^{\prime}(x)>0 \text { for } x<3} \\\ {f^{\prime}(3) \text { does not exist. }} \\ {f^{\prime}(x)<0 \text { for } x>3} \\ {f^{\prime \prime}(x)>0, x \neq 3}\end{array} $$

Step-by-Step Solution

Answer

The function \(f\) passes through the points \(x=2\) and \(x=4\), increases and is concave up for \(x<3\), has a sharp turn at \(x=3\), and is decreasing and concave up for \(x>3\).

Step 1: Sketch a Preliminary Graph from the Zeros of the Function

We can begin sketching the graph by marking the points where the function equals 0, which are \(x=2\) and \(x=4\). Draw a horizontal x-axis and mark these points.

Step 2: Determine the Increasing and Decreasing Intervals

According to the problem, the derivative of the function (which dictates the slope of the graph) is positive for \(x<3\) and negative for \(x>3\). This means that the function should be increasing for \(x<3\) and decreasing for \(x>3\) since a positive derivative indicates an increasing function while a negative derivative indicates a decreasing function.

Step 3: Indicate the Point of Non-Differentiability on the Graph

The problem also indicates that \(f'(3)\) doesn't exist. This means there must be a sharp turn in the graph at \(x=3\). We can thus draw a vertical dotted line at \(x=3\) and mark this point.

Step 4: Determine the Concavity of the Function Sections

Finally, the second derivative of the function is stated to be positive for all \(x\) not equal to 3. This means that the graph is concave up, or 'smiling,' on these intervals (since a positive second derivative signifies a concave up function).

Step 5: Combine all the Information to Complete the Sketch

Bringing all the information above together, the graph should pass through \(x=2\) and \(x=4\) on the x-axis, be increasing and concave up on the interval \(x<3\), have a sharp turn at \(x=3\), and be decreasing and concave up for \(x>3\).

Key Concepts

Increasing and Decreasing Intervals

Understanding the increasing and decreasing behavior of a function is crucial when it comes to sketching its graph. Simply put, if a function is increasing, its graph goes upwards as you move from left to right. On the other hand, if the function is decreasing, the graph descends in the same direction. But how do we determine whether a function is increasing or decreasing? We look at the sign of the first derivative, denoted as \(f'(x)\).

When \(f'(x) > 0\), the function is increasing; this means that as \(x\) becomes larger, so does \(f(x)\). Conversely, if \(f'(x) < 0\), the function is decreasing, meaning that \(f(x)\) becomes smaller as \(x\) increases. In the given exercise, \(f'(x) > 0\) when \(x < 3\) and \(f'(x) < 0\) when \(x > 3\). As a result, the graph of \(f\) must be increasing on the interval to the left of \(x=3\) and decreasing to the right of \(x=3\).

This concept lays the foundation for predicting the general shape of a function's graph before we even start plotting points or considering concavity. It is an integral step in sketching any function.

Point of Non-Differentiability

In the realm of calculus, the differentiability of a function at a point is a sign of how 'smooth' the graph is at that location. A function is considered non-differentiable at a point if the graph has a sharp corner, a vertical tangent, or a discontinuity there. The exercise hints at the presence of such a point through the condition \(f'(3)\) does not exist. This tells us that at \(x=3\), the graph of \(f\) cannot be smoothly traced with a single, flowing movement of the pencil – there’s an abrupt change in direction.

One common type of non-differentiable point is a 'cusp' or a sharp turn, as indicated in our step-by-step problem solution. When graphing, we can represent this by drawing a dotted vertical line to suggest that the derivative doesn't exist at this point. Recognizing points of non-differentiability is essential when sketching functions because it often indicates a significant feature in the graph that we can't miss. By marking these points, we give a clearer picture of the behavior and shape of the function.

Concavity and Second Derivative

Concavity and the second derivative of a function are closely tied together. They give us information about the 'bend' of the graph. When the second derivative \(f''(x) > 0\), the function is concave up, resembling the shape of a smile or an upward-facing bowl. Alternatively, when \(f''(x) < 0\), the function is concave down, similar to a frown or a downward-facing bowl.

In the problem provided, \(f''(x) > 0\) for all \(x\) except at \(x = 3\). This tells us that, apart from at \(x = 3\), the graph of \(f\) should have a concave up shape. Interestingly, concavity can give us deeper insights into a function's behavior – concave up intervals often imply that the rate of increase is itself increasing, while concave down intervals suggest the increase rate is decreasing or the decrease rate is increasing.

To visually represent concavity on a graph, consider using a curved line that opens upwards for concave up sections and downwards for concave down sections. This concept helps with predicting the function’s behavior between critical points and inflection points which is significant for accurately rendering the graph’s curvature.