Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=3 ; f(3)=0 ; f(6)=4\); $$ \begin{aligned} &f^{\prime}(x)<0 \text { on }(0,3) ; f^{\prime}(x)>0 \text { on }(3,6) \\ &f^{\prime \prime}(x)>0 \text { on }(0,5) ; f^{\prime \prime}(x)<0 \text { on }(5,6) \end{aligned} $$

Step-by-Step Solution

Answer

The function decreases on \((0,3)\), shifts from concave up to concave down at \(x=5\), and increases on \((3,6)\). Sketch reflects these traits.

Step 1: Analyze given conditions

The given conditions for the function are:- It is continuous on \([0,6]\) and has defined values at three key points: \(f(0) = 3\), \(f(3) = 0\), and \(f(6) = 4\).- The derivative \(f'(x) < 0\) on \((0,3)\) indicates the function is decreasing.- The derivative \(f'(x) > 0\) on \((3,6)\) indicates the function is increasing.- The second derivative \(f''(x) > 0\) on \((0,5)\) indicates the function is concave up, resembling a bowl shape.- The second derivative \(f''(x) < 0\) on \((5,6)\) implies the function is concave down, resembling an upside-down bowl.

Step 2: Sketch segments according to the derivative signs

Start from \(f(0) = 3\):1. From \(0\) to \(3\), since \(f'(x) < 0\), the function decreases from \(3\) down to \(0\) at \(x = 3\). This section should be drawn as decreasing and concave up (\(f''(x) > 0\)).2. From \(3\) to \(6\), since \(f'(x) > 0\), the function increases from \(0\) to \(4\) at \(x = 6\). This section needs to be increasing and concave up until \(x = 5\), then concave down from \(x = 5\) to \(x = 6\).

Step 3: Identify inflection point

There is a change in concavity at \(x = 5\) since \(f''(x)\) changes from positive to negative. This indicates an inflection point at \(x = 5\). Ensure the sketch reflects the change from concave up to concave down at this point. The function continues increasing through this change until it reaches \(f(6) = 4\).

Step 4: Connect the points and shape properly

Connect the segments ensuring smoothness and continuity:- Between \(f(0) = 3\) and \(f(3) = 0\), the curve should be a downward concave-up shape.- Between \(f(3) = 0\) and \(f(6) = 4\), it should start concave up as it goes upward, and then switch to concave down at \(x = 5\) as it continues to \(f(6) = 4\).This will ensure all function and derivative conditions are met.

Key Concepts

Derivatives

Understanding derivatives is key to sketching graphs of continuous functions. Derivatives can give us insight into the behavior of a function at various intervals. In this exercise, the first derivative, denoted as \( f'(x) \), provides information about the slope of the function:

• If \( f'(x) < 0 \), it indicates the function is decreasing. Here, this happens when \( x \) is between 0 and 3.
• If \( f'(x) > 0 \), the function is increasing, as seen between 3 and 6 on the x-axis.

The derivative thus helps in identifying whether sections of the graph should move up or down as you proceed along the x-axis. Graphing these trends is essential for shaping the curve of the continuous function accurately. By analyzing these intervals, you can effectively predict and sketch the progression of the function through its domain.

Concavity

Concavity relates to how a function bends along its domain, giving us an idea about its curvature. This is described using the second derivative of the function, noted as \( f''(x) \). Here's how the second derivative relates to concavity:

• When \( f''(x) > 0 \), the function is concave up, resembling a U-shape or bowl. In this exercise, the function is concave up from \( x = 0 \) to \( x = 5 \).
• When \( f''(x) < 0 \), it becomes concave down, akin to an upside-down bowl. This occurs between \( x = 5 \) and \( x = 6 \).

By looking to these shifts in concavity, we foresee moments where the graph will change its bending direction. This insight helps in creating a more accurately sketched function and understanding how it maneuvers through different regions.

Inflection Points

Inflection points are significant points on a graph where the concavity changes from up to down or vice versa. Identifying these helps us recognize pivotal shifts in the graph's curvature. In the given problem, an inflection point exists at \( x=5 \). Here, \( f''(x) \) flips its sign from positive to negative, indicating the transition from concave up to concave down.The identification of inflection points is critical for ensuring that your graph reflects an accurate transition in the curve’s behavior. By marking these points, you ensure the graph's curvature aligns with the given conditions, providing a more comprehensive and detailed representation of the function.