Question

Sketch the graph of a function that is continuous on an open interval \((a, b)\) but has neither an absolute maximum nor an absolute minimum value on \((a, b)\)

Short answer

Question: Provide an example of a function that is continuous on an open interval (a, b) but has neither an absolute maximum nor an absolute minimum value on (a, b). Answer: An example of such a function is the sine function, f(x) = sin(x), on the open interval (a, b). This function is continuous and periodic, and it has no absolute maximum or minimum values within this interval.

Step-by-step solution

Identify an appropriate function

To satisfy the required conditions, we can choose a simple sinusoidal function, like the sine function, which is continuous everywhere but has no absolute maximum or minimum values in general (unless it's restricted to a closed interval). We'll use the sine function on the open interval \((a, b)\), which can be defined as: $$ f(x) = \sin(x) $$

Sketch the sine function

We'll sketch the sine function on the open interval \((a, b)\). Draw the x-axis and y-axis. Since the sine function is periodic with a period of \(2\pi\), we need to determine the number of full periods within the interval \((a, b)\). We'll consider a general case, where there are n full periods in \((a, b)\). To do this, ensure that the sketch includes the characteristic sinusoidal wave pattern between the points \((a, b)\). Make sure the waveform starts at \(\sin(a)\) and ends at \(\sin(b)\).

Indicate the absences of absolute maximum and minimum values

Add a note to your sketch indicating that there are no absolute maximum and minimum values. You can mention that since the sine function is both continuous and periodic, it has no absolute maximum or minimum values on the open interval \((a, b)\). The final sketch should represent the sine function bounded by the open interval \((a, b)\) and show the characteristic wave pattern. Additionally, the sketch should indicate the absence of absolute maximum and minimum values within the interval.

Key concepts

Sinusoidal Function

A sinusoidal function is one of the most important forms of periodic functions frequently encountered in various branches of mathematics and physics. It is characterized by its smooth, wave-like pattern, which can be described using either the sine or cosine functions. When you think of a sinusoid, imagine the undulating waves of the ocean or the vibrations of a guitar string.

The general form of a sinusoidal function is \( y = A\sin(B(x - C)) + D \) or \( y = A\cos(B(x - C)) + D \) where:\

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• \(A\) represents the amplitude, the peak deviation of the function from zero.\
• \(B\) affects the period of the function, which is the distance over which the function's shape repeats.\
• \(C\) is the horizontal shift, determining where the wave starts.\
• \(D\) is the vertical shift, setting the midline of the wave.\

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With sinusoidal functions, even though they undulate infinitely, if defined over an open interval such as \( (a, b) \), they do not possess an absolute highest (maximum) or lowest (minimum) point. They continuously rise and fall in a predictable, periodic manner, making them both elegant and predictable within their bounds.

Periodic Functions

Periodic functions are the heartbeats of mathematics, repeating their values at regular intervals or periods. This consistency means that once you know the function over one interval, you effectively understand its behavior everywhere. The periodicity of a function is the length of the smallest interval that contains one full pattern of repetition.

\Example of Periodicity\\For the sine function, which is an exemplar of periodic behavior, the base period is \( 2\pi \), corresponding to a full cycle of the wave. Practically, this means that \( \sin(x) = \sin(x + 2\pi) \) for all values of \( x \), and similarly for any integer multiple of \( 2\pi \). When sketching periodic functions on an open interval, it's crucial to capture the essence of this repeating pattern, showing only a portion of its infinite extension, without suggesting the existence of ultimate peaks or troughs.

Absolute Extrema

In the realm of calculus, the terms 'absolute extrema' refer to the highest (maximum) and lowest (minimum) values that a function can achieve over a certain interval. Put simply, these are the peak and valley points where the function doesn't get any higher or any lower respectively.

When restricted to a closed interval, a continuous function must encompass both an absolute maximum and an absolute minimum due to the Extreme Value Theorem. However, when we cast our gaze over an open interval, like \( (a, b) \), the function is no longer bound to these rules. It can flirt with extrema without ever making the relationship 'absolute'. This elusive behavior is inherent to functions like the sine wave, which oscillates continuously without ever settling on a single highest or lowest point within the period—as long as the interval remains open. Hence, these functions can be continuous and bounded, yet forever undulate between their curved confines.