Problem 2 - Calculus - AP Edition

Question

In each part, sketch the graph of a continuous function \(f\) with the stated properties. (a) \(f\) has exactly one relative extremum on \((-\infty,+\infty)\) and \(f(x) \rightarrow 0\) as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). (b) \(f\) has exactly two relative extrema on \((-\infty,+\infty)\) and \(f(x) \rightarrow 0\) as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). (c) \(f\) has exactly one inflection point and one relative extremum on \((-\infty,+\infty)\). (d) \(f\) has infinitely many relative extrema, and \(f(x) \rightarrow 0\) as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\).

Step-by-Step Solution

Answer

Use specific polynomial or trigonometric transformations to match the conditions for each part, noting extrema and end behavior.

Step 1: Understanding the Exercise

In this exercise, we need to sketch graphs of continuous functions with certain characteristics. For each part, we'll define what the function should look like based on the properties given.

Step 2: Part (a) - One Relative Extrema and Limits to Zero

For part (a), the function \( f \) has exactly one relative extremum. This means it has either a maximum or minimum point. The behavior as \( x \to \pm\infty \) implies the function should flatten out towards the x-axis. A simple example is an upside-down or regular parabola, like \( f(x) = -x^2 + 1 \), which has a maximum, or \( f(x) = -\frac{x^2}{x^2+1} \), where \(f(x)\to 0 \) as \( x \to \pm\infty \).

Step 3: Part (b) - Two Relative Extrema and Limits to Zero

For part (b), \( f \) has two relative extrema, a minimum and a maximum, so we look for a function like \( f(x) = x^3 - 3x \), which has two turning points and as \( x \rightarrow \pm\infty \), \( f(x) \rightarrow \infty \), not zero. Instead, use something like \( f(x) = \frac{x^3 - 3x}{x^2 + 1} \) to have \( f(x) \rightarrow 0 \).

Step 4: Part (c) - One Inflection Point and One Relative Extrema

For part (c), the function \( f \) should have an inflection point and one relative extremum. An inflection point is where the concavity changes. The cubic function \( f(x) = x^3 - 3x \) has a point of inflection and a stationary point. We can modify it to \( f(x) = \frac{x^3 - 3x}{x^2+1} \) to ensure it approaches zero at infinity.

Step 5: Part (d) - Infinitely Many Relative Extrema and Limits to Zero

For part (d), the function \( f \) needs infinitely many extrema and tends to zero as \( x \to \pm\infty \). A trigonometric form such as \( f(x) = \frac{\sin(x)}{x} \) fits, as it oscillates with infinitely many peaks and valleys getting closer to zero at infinity.

Key Concepts

Relative Extrema

In calculus, relative extrema are important features of a function's graph. These points are where the function changes direction, signaling either a peak or a valley in the graph.

• A relative maximum occurs when a function changes from increasing to decreasing.
• A relative minimum happens when it changes from decreasing to increasing.

These can be identified using the first derivative test. When the derivative of a function, represented as \(f'(x)\), equals zero, a potential extremum is found. In the exercise, depending on whether one or two relative extrema are present, the function might look like a simple parabola or more complex cubic functions.It's fascinating to see how functions with multiple extremum reflect multiple turns, showing complexity in their shapes. Understanding these concepts is key, as they help in predicting how a function behaves and sketching its graph accurately.

Inflection Point

An inflection point on a function's graph is where the curvature changes direction. This means the graph shifts from being concave upwards to concave downwards, or vice versa.

• This occurs where the second derivative, \(f''(x)\), changes sign.
• It indicates a change in the rate of change of the slope, adding an interesting twist to the graph's shape.

In the exercise, having both an inflection point and a relative extremum makes the graph even more dynamic. The inflection point allows us to visualize "turning" beyond just peaks and valleys, offering a fuller picture of the function's behavior across its domain.

Limits at Infinity

Understanding limits at infinity helps us sketch functions correctly as they extend towards extreme values of \(x\). This concept refers to the behavior of a function as \(x\) approaches positive or negative infinity.

• If \(f(x) \rightarrow 0\), the function is leveling off, getting closer to the x-axis as \(x\) becomes very large or very small.
• This was crucial in these exercises, where the desired behavior was for functions to approach zero as \(x\) tended towards either infinity.

Understanding this concept helps with predicting and sketching the long-term behavior of the graph of a function. Checking these infinite limits ensures that our sketch reflects how the function behaves not just between marked points, but as it continues indefinitely in both directions.