Problem 96 - Thomas Calculus

Question

Sketch a smooth connected curve \(y=f(x)\) with \begin{equation}\begin{array}{ll}{f(-2)=8,} & \quad \quad \quad \quad \quad \quad {f^{\prime}(2)=f^{\prime}(-2)=0,} \\ {f(0)=4,} & \quad \quad \quad \quad \quad \quad {f^{\prime}(x)<0} {\text { for } \quad|x|<2,} \\ {f(2)=0,} & \quad \quad \quad \quad \quad \quad {f^{\prime \prime}(x)<0} {\text { for } x<0,} \\ {f^{\prime}(x)>0} & {\text { for }|x|>2, \quad f^{\prime \prime}(x)>0 \quad \text { for } \quad x>0,}\end{array}\end{equation}

Step-by-Step Solution

Answer

Sketch a curve decreasing from \((-2, 8)\) to \((2, 0)\), changing concavity at \(x=0\), showing a local maximum at \((-2, 8)\).

Step 1: Analyze Points and Critical Values

The curve must pass through specific points: \((-2, 8)\), \((0, 4)\), and \((2, 0)\). We know \(f'(2) = f'(-2) = 0\), indicating potential local extremum points at \(x = -2\) and \(x = 2\). Since these are the points where the derivative is zero, they are likely candidates for turning points where the slope changes from positive to negative or vice versa.

Step 2: Determine Derivative Behavior Regions

The first derivative \(f'(x)<0\) when \(|x|<2\) tells us that the function is decreasing in the region around the origin from \(-2 < x < 2\). When \(|x|>2\), \(f'(x)>0\) implies the function is increasing in those outer regions. This means the curve is falling as it approaches \(x = -2\) from the right and rising again past \(x = 2\).

Step 3: Analyze Concavity

For the second derivative, \(f''(x)<0\) when \(x<0\) indicates concave down in the left half of the plane. This means that in this region, the function may have a maximum. Similarly, \(f''(x)>0\) for \(x>0\) tells us it's concave up, implying a local minimum could be at \(x=0\). This \(x=0\) point with \(f(0)=4\) is likely where the curve turns concave up.

Step 4: Sketch the Curve

Start by plotting key points: \((-2, 8)\), \((0, 4)\), and \((2, 0)\). Sketch a curve that descends from a high point at \((-2, 8)\), decreases to \((0, 4)\) while being concave down until it switches to concave up, then continues decreasing to \((2, 0)\) where it begins to increase again. The curve should be smooth with no breaks, following the given derivative conditions to complete the sketch effectively.

Key Concepts

Critical Points

Critical points are essential features of a curve and include where a function's derivative equals zero or is undefined. In our exercise, we find these critical points at \(x = -2\) and \(x = 2\), which are given since \(f'(x) = 0\) at these points.
Critical points can signal where the graph might have a peak or valley. These are essential in determining where to best sketch changes in direction on the curve.
With values like \(f(-2) = 8\) and \(f(2) = 0\), we understand that these points can potentially be a local maximum or minimum. Additionally, marking these points helps guide the flow of the entire graph when combined with derivative behavior.
As we sketch, it's crucial to recognize that at critical points, the slope is neither rising nor falling but momentarily flat. This information helps plot the pivotal parts of the function's graph.

Derivative Behavior

The derivative behavior of a function indicates how the function is changing at each interval.
For our curve, the first derivative \(f'(x)\) reveals:

• When \(|x| < 2\), \(f'(x) < 0\), meaning the function is decreasing.
• Conversely, when \(|x| > 2\), \(f'(x) > 0\), implying that the function is increasing.

This behavior helps us visualize the curve differently in distinct intervals. Between \(x = -2\) and \(x = 2\), where the function decreases, the graph is moving downwards.
Outside this interval, the graph moves upwards, indicating increasing behavior. Knowing how these intervals behave is key in accurately determining the graph’s shape and where peaks and troughs might exist.

Concavity

Concavity tells us the direction the curve bends over each interval, defined by the second derivative \(f''(x)\). For our function, the details are:

• For \(x < 0\), \(f''(x) < 0\), showing the function is concave down.
• For \(x > 0\), \(f''(x) > 0\), indicating it is concave up.

This characteristic suggests where inflection points might occur and helps in determining the nature of local extrema. The switch from concave down to concave up at \(x=0\) demonstrates a possible turning point where the graph could transition through a minimum.Using concavity alongside derivative behavior modifies how the sketch is drawn, by smoothly connecting and transitioning the curve across these turning points.

Local Extremum

Local extrema refer to points where the graph reaches a local maximum or minimum. Given the derivative conditions, we can estimate these in our exercise.
With \(f'(x) = 0\) at \(x = -2\) and \(x = 2\), combined with the \(f(x)\) values, these points serve as potential local extrema.

• At \(x = -2\), with \(f(-2) = 8\), there might be a local maximum since the graph decreases after this point.
• At \(x = 2\), where \(f(2) = 0\), we find a local minimum because the graph rises following this point.
• The concavity details further aid in this determination.

Understanding the extremum provides insight into how to best connect the curve across critical points. It helps us visualize and anticipate the curve's highest and lowest points, reinforcing the sketch with accurate details.