Problem 4 - Calculus Early Transcendentals

Question

In each part, sketch the graph of a continuous function \(f\) with the stated properties on the interval \((-\infty,+\infty)\) (a) \(f\) has no relative extrema or absolute extrema. (b) \(f\) has an absolute minimum at \(x=0\) but no absolute maximum. (c) \(f\) has an absolute maximum at \(x=-5\) and an absolute minimum at \(x=5\).

Step-by-Step Solution

Answer

(a) Linear, (b) Quadratic up, (c) Parabolic upside-down.

Step 1: Understanding Problem (a)

For part (a), we need to sketch the graph of a function \( f \) that neither rises to a peak nor falls to a trough, meaning no relative minima or maxima, and it continues infinitely without reaching a highest or lowest point globally.

Step 2: Sketch for (a): Linear Function

An example of such a function is \( f(x) = x \). This is a straight line with a constant slope, increasing or decreasing uniformly, without any peaks or troughs, satisfying the condition of no relative or absolute extrema.

Step 3: Understanding Problem (b)

For part (b), the function \( f \) should reach its lowest point at \( x=0 \), and continue to increase indefinitely as \( x \to \infty \) without reaching a highest point.

Step 4: Sketch for (b): Quadratic Function Opening Up

A suitable function is \( f(x) = x^2 \) with its vertex, the absolute minimum, at \( x = 0 \). The function increases in both directions away from the vertex, thus having no absolute maximum as \( x \to \pm \infty \).

Step 5: Understanding Problem (c)

In part (c), \( f(x) \) must achieve the highest point at \( x=-5 \) and the lowest at \( x=5 \). Beyond these points, the function should not exceed these extremes.

Step 6: Sketch for (c): Parabolic Function Upside-Down

Consider \( f(x) = -(x+5)^2 + 25 \) for \( x < 5 \) and \( x > -5 \), with a transformation causing symmetry around these points. The function decreases rain-like from \( x=-5 \) to \( x=5 \), fulfilling the extremum conditions.

Key Concepts

Continuous Functions

A continuous function is a fundamental concept in calculus. It means that the graph of function has no breaks, holes, or jumps, making it "smooth" and unbroken across an interval.
This feature ensures that you can draw the function without lifting your pencil from the paper. When describing continuous functions mathematically, use the concept of limit:
A function \( f(x) \) is continuous at a point \( c \) if \( \lim_{x \to c} f(x) = f(c) \).
This definition means that as \( x \) gets arbitrarily close to \( c \), \( f(x) \) approaches \( f(c) \).

• No abrupt changes in direction.
• No undefined points, where the function cannot be determined.
• Smooth connections between values.

Employing continuous functions is crucial for ensuring that calculations perform as expected across an input range.

Relative Extrema

Relative extrema, which include relative (or local) minimums and maximums, are important points on a graph where the function changes direction from increasing to decreasing or vice versa.
At a relative maximum, the function has a peak, and at a relative minimum, it has a trough. To find these points, utilize the first derivative test:
Ensure that you take the first derivative of the function \( f'(x) \), and check where \( f'(x) = 0 \) or is undefined.
These critical points are candidates for relative extrema.

• If the sign of \( f'(x) \) changes from positive to negative, there is a relative maximum.
• If it changes from negative to positive, there is a relative minimum.

Relative extrema are key in understanding the local behavior of a curve, indicating the rise and fall patterns within a specific domain.

Absolute Extrema

Absolute extrema are the ultimate highs and lows across the entire domain of a function, which includes both finite and infinite intervals. They define the top and bottom values a function can take.
If a function reaches its absolute maximum at \( x = a \), then \( f(a) \) is the highest value within all \( x \) in the domain. Similarly, the absolute minimum is the lowest point.
Determining absolute extrema requires evaluating the function at critical points and the endpoints of the interval (if they exist).

• Absolute extrema provide a global perspective on the function.
• Unlike relative extrema, they consider the entire range of the function.
• The extrema are applied in optimization problems to find optimal solutions.

In mathematical optimization or business applications, these points are pivotal for finding efficient and optimal solutions.

Graph Sketching

Graph sketching is a skill that helps visualizing the behavior of functions. It involves drawing a quick, rough representation of the function's graph that captures essential features such as intercepts, axis crossings, and local and global turns.
Here are some vital steps to sketch a graph:

• Identify key points like intercepts, zeros, and any given critical points.
• Analyze the function's first and second derivatives to determine relative extrema and concavity.
• Note end behavior, as \( x \to \pm \infty \), to understand how the function stretches or compresses beyond extremes.

Through graph sketching, interpreting function features becomes intuitive, making it a valuable tool for efficiently understanding equations and their implications.