Problem 3 - Calculus - AP Edition

Question

In each part, sketch the graph of a continuous function \(f\) with the stated properties on the interval [0,10] (a) \(f\) has an absolute minimum at \(x=0\) and an absolute maximum at \(x=10\) (b) \(f\) has an absolute minimum at \(x=2\) and an absolute maximum at \(x=7\) (c) \(f\) has relative minima at \(x=1\) and \(x=8,\) has relative maxima at \(x=3\) and \(x=7,\) has an absolute minimum at \(x=5,\) and has an absolute maximum at \(x=10\)

Step-by-Step Solution

Answer

Graphs show increasing curves for (a), a U-shaped curve peaking at 7 in (b), and a wavy curve with highs and lows in (c).

Step 1: Analyze Part (a)

For part (a), the function \(f\) is continuous on the interval \([0, 10]\) and it has an absolute minimum at \(x=0\) and an absolute maximum at \(x=10\). This means the function starts at its lowest point and ends at its highest point.

Step 2: Sketch Graph for Part (a)

Draw the interval from \(x=0\) to \(x=10\) on the x-axis. Plot a point at \(x=0\) with a low y-value, the absolute minimum. Plot another point at \(x=10\) with a higher y-value, the absolute maximum. Connect these two points with a smooth, continuous curve that could resemble an increasing function.

Step 3: Analyze Part (b)

In part (b), the function \(f\) on \([0, 10]\) has an absolute minimum at \(x=2\) and an absolute maximum at \(x=7\). This suggests that the graph reaches its lowest point at \(x=2\) and highest point at \(x=7\).

Step 4: Sketch Graph for Part (b)

Begin plotting at \(x=0\) and draw towards \(x=2\), where the absolute minimum occurs. From \(x=2\), increase to reach \(x=7\) at the absolute maximum. Then, draw from \(x=7\) to \(x=10\), ensuring the graph remains smooth and continuous.

Step 5: Analyze Part (c)

For part (c), the function is continuous and has relative minima at \(x=1\) and \(x=8\), relative maxima at \(x=3\) and \(x=7\), an absolute minimum at \(x=5\), and an absolute maximum at \(x=10\). We must ensure these points are represented as peaks or valleys as specified.

Step 6: Sketch Graph for Part (c)

Begin the graph at \(x=0\). Approach \(x=1\) for a relative minimum and rise to \(x=3\) for a maximum. Continue to \(x=5\) for the absolute minimum, then rise to \(x=7\) for a second relative maximum. Drop to \(x=8\) for another relative minimum and end at \(x=10\) at the absolute maximum. Connect all points with a smooth line to form a continuous graph that fits the descriptions.

Key Concepts

Absolute Minimum

An absolute minimum is the lowest point that a function reaches on a given interval. It represents the smallest y-value of the function across its domain. When we say a function has an absolute minimum at a certain point, this means that the function attains its minimum value there, no part of the function for the specified interval goes lower than this point.

Here’s what to remember about absolute minimum:

• The function can have only one absolute minimum in a specified interval. However, it can have more than one absolute minimum value if the function value is the same at two or more points. In this exercise, the function has an absolute minimum at specified points like at (0, f(0)) for part (a), where the function begins at its lowest point.
• In another illustration, part (b) describes an absolute minimum at (2, f(2)).

To visualize this on a graph, think of it like a valley or the bottom of a U-shape, a point lower than any other point on the interval.

Absolute Maximum

The absolute maximum of a function is the highest point it reaches over a certain interval. It’s the point where the function's y-value is the largest, meaning no other points in that range surpass it in height.

Key points to understand absolute maximum include:

• Similar to an absolute minimum, a function can only have one absolute maximum over a specified interval, unless the same maximum value is repeated over different points. For part (a) of the exercise, the function reaches its absolute maximum at (10, f(10)), which is the highest point in the given interval [0, 10].
• In part (b), the absolute maximum is at (7, f(7)), indicating the pinnacle of the graph over that interval.

When sketching the graph, you'll represent an absolute maximum typically at the peak of a curve or as the endpoint of an increasing function.

Relative Minimum

A relative minimum refers to a point on the graph of a function that is lower than all nearby points but may not be the lowest overall point in the interval. This means it’s smaller than its immediate surroundings, but there may be other relative or absolute minimum points elsewhere.

Think of a relative minimum like a small dip or valley on the graph:

• For part (c) of the exercise, the function exhibits relative minima at points (1, f(1)) and (8, f(8)). These provide localized low points which can be thought of as minor depressions in the curve, not as deep as the overall lowest point of the interval.
• It's important for a function to increase as it approaches from both sides of a relative minimum, creating that distinct "U" shape around the graph at that point.

This concept helps to visualize peaks and troughs which might not be the prominent points but create essential dynamics in the graph’s structure.

Relative Maximum

A relative maximum is a point on a function's graph where it reaches a higher value than all immediately surrounding points. However, it is not necessarily the highest point on the entire graph (which would be an absolute maximum). A relative maximum provides a local peak within the interval.

Here are the core ideas about relative maximum:

• For exercise part (c), the function has relative maxima at (3, f(3)) and (7, f(7)). This indicates peaks that are higher than nearby points but might not be the overall highest points in the function's domain.
• Just as with relative minima, relative maxima require the function to decrease on either side of the peak to form a discernible peak or "hill."

To think about this visually, imagine a gentle hill or a bump rising above surrounding valleys. A function can have multiple relative maxima, and they contribute to the function’s undulating form across the graph.