Problem 2 - Calculus - AP Edition

Question

In each part, sketch the graph of a function \(f\) with the stated properties. (a) \(f\) is increasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave up on \((0,+\infty)\) (b) \(f\) is increasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave down on \((0,+\infty)\) (c) \(f\) is decreasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave up on \((0,+\infty)\) (d) \(f\) is decreasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave down on \((0,+\infty)\)

Step-by-Step Solution

Answer

(a) Use \(f(x) = x^3\); (b) Use \(f(x) = x^3 - x\); (c) Use \(f(x) = -x^3 + x\); (d) Use \(f(x) = -x^3\).

Step 1a: Sketch for Part (a)

To satisfy the conditions of part (a), we need a function that is always increasing. A suitable example is a cubic function. Consider the function \(f(x) = x^3\). This function is increasing everywhere since its derivative \(f'(x) = 3x^2\) is positive for all \(x\). The second derivative, \(f''(x) = 6x\), is zero at \(x = 0\), indicating an inflection point. Furthermore, the second derivative is positive for \(x > 0\) (i.e., \(f''(x) > 0\)), confirming that the function is concave up on \((0, +\infty)\). Thus, \(f(x) = x^3\) satisfies all given conditions.

Step 1b: Sketch for Part (b)

To meet the conditions of part (b), select a function that increases throughout the real number line and has an inflection point at the origin, but is concave down on \((0, +\infty)\). Consider the function \(f(x) = -x^3\). The derivative \(f'(x) = -3x^2\) is negative for all \(x\), which mistakenly indicates a decreasing function; hence, use \(f(x) = x^3 - x\). Here, \(f'(x) = 3x^2 - 1\) is positive for \(x \in (-\infty, +\infty)\), and \(f''(x) = 6x\) changes from negative to positive at the origin. Additionally, the second derivative is negative for \((0, +\infty)\), showing concavity shifts as required.

Step 2a: Sketch for Part (c)

For part (c), the function must be decreasing over all \(x\) and have the characteristics of concave up on \((0, +\infty)\). Use \(f(x) = -x^3\). This choice works because \(f'(x) = -3x^2\) is always negative, confirming the function decreases everywhere. The inflection point at the origin comes from \(f''(x) = -6x\), which is zero at \(x = 0\). For \(x > 0\), \(f''(x) < 0\), suggesting that part of the original allocation to \(f(x) = x^3\) is instead \(f(x) = -x^3 + x\), for which \(f'(x) = -3x^2 + 1\) and \(f''(x) = 6x - 1\) demonstrate concavity changes.

Step 2b: Sketch for Part (d)

In part (d), the function must decrease everywhere and be concave down on \((0, +\infty)\). Although \(f(x) = -x^3\) fully aligns with these criteria: it decreases on the whole real line, as shown by \(f'(x) = -3x^2\). Its second derivative \(f''(x) = -6x\) confirms an inflection point at the origin and shows it is concave down for \(x > 0\) since \(f''(x) < 0\) there.

Key Concepts

Function Graph Sketching

When sketching the graph of a function, several key details and properties need to be considered to accurately visualize the function's behavior.

• **Function Definition**: Begin by understanding basic features such as whether the function is increasing or decreasing across its domain.
• **Observing Growth Trends**: For instance, functions like cubic functions can be either strictly increasing or decreasing. This will guide the overall shape of the graph.
• **Critical Points**: Identify critical points where the derivative is zero or undefined; they mark potential maxima, minima, or inflection points where the concavity changes.

For each exercise case, considering a known function model like a cubic equation \( f(x) = x^3 \) or \( f(x) = -x^3 \) simplifies sketching when aligning these models to provided conditions, ensuring continuity and correctly depicting entire behavior across given intervals.

Inflection Points

Inflection points are where the curve changes its concavity, transitioning from concave up to concave down, or vice versa.

• **Calculation Method**: Calculated using the second derivative test, where the second derivative \( f''(x) \) changes signs at the inflection point.
• **Graphical Implication**: This is a point on the graph with a noticeable change in the curvature.

For a function with an inflection point at the origin, as suggested by exercises, the requirement is that \( f''(x) = 0 \) exactly at \( x = 0 \) and that it switches its sign (from positive to negative or vice versa) around that value.
Recognizing where these points lie helps in sketching the function more realistically and in alignment with calculus principles.

Concavity Analysis

Concavity tells us the direction in which a function curves. This can be categorized into:

• **Concave Up**: When the graph forms a 'cup' shape and its second derivative \( f''(x) > 0 \); the tangent line sits below the curve.
• **Concave Down**: Where the graph forms a 'cap' shape and its second derivative \( f''(x) < 0 \); the tangent line runs above the curve.

Proper concavity analysis is crucial when sketching graphs since it affects the overall shape and helps pinpoint inflection points.
To ensure precision, you calculate and observe the sign of the second derivative function over different intervals which specify transitions accurately, ensuring the graph bends correctly on each specified domain of the function.

Derivative Testing

Derivative testing involves calculating the first and second derivatives of a function to analyze its growth or decline and curvature.

• **First Derivative\( f'(x) \) Testing**: Dictates whether a function is increasing (\( f'(x) > 0 \)) or decreasing (\( f'(x) < 0 \)) over specific intervals.
• **Second Derivative\( f''(x) \) Testing**: Informs about the concavity and helps locate inflection points.

Deploying these tests systematically allows students to approach function graph sketching methodically.
Ensuring functions meet the given conditions: like being consistently increasing or decreasing, and accurately capturing the behavior across all necessary intervals, helps in achieving a focused function sketching plan aligned with calculus fundamentals.