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To identify increasing intervals of a function, we examine where its derivative is greater than zero. For the function in this exercise, given by \( f(x) = 1 - \ln x \), we find the derivative as \( f'(x) = -\frac{1}{x} \). Since \( f'(x) = -\frac{1}{x} \) is always negative for \( x > 0 \), there are no values of \( x \) where the derivative is positive.

This means that in no interval is the function \( f(x) \) increasing. It is important to understand that if the derivative were positive, it would indicate an increasing interval. However, since it is completely negative across its entire domain \( (0, \infty) \), \( f \) is not increasing anywhere.

Understanding increasing intervals can help in sketching graphs as they show where the function goes up as \( x \) increases.

Decreasing intervals occur where the derivative of a function is less than zero. For our function, \( f(x) = 1 - \ln x \), the derivative is \( f'(x) = -\frac{1}{x} \). In this case, \( f'(x) \) is negative for all \( x > 0 \). This consistent negative value indicates that \( f(x) \) is decreasing over its entire domain, which is from 0 to infinity \((0, \infty)\).

Visualizing a decreasing graph helps students see how the function goes down as \( x \) increases. This continuous decrease means that at each point in its domain, the values of \( f(x) \) will always be less than the values preceding them, illustrating a perfect downhill path on a graph.

An essential takeaway is that negative derivatives over specific intervals signal where the function declines steadily.

Finding relative maxima and minima involves identifying points where the derivative equals zero or is undefined, and then evaluating whether these points are peaks (maxima) or valleys (minima). For \(f(x) = 1 - \ln x\), we calculated \(f'(x) = -\frac{1}{x}\).

Interestingly, \(f'(x)\) never equals zero for any positive \(x\), so there are no relative maxima or minima. Additionally, \(f'(x)\) is undefined at \(x = 0\), yet this doesn't contribute to a critical point within the domain \( (0, \infty)\).

With no locations where the slope of the function levels off or switches directions, we conclude the function has no high or low points in this traditional sense. This highlights the critical role played by the derivative in detecting and confirming these extreme values.

Always look for zero or undefined derivatives to explore potential extremities.

Graphing a function effectively portrays its behavior over different intervals. For the function \( f(x) = 1 - \ln x \), let's start by understanding its general shape. As \( x \to 0^+ \), \( \ln x \to -\infty \), meaning \( f(x) \to \infty \). As \( x \to \infty \), \( \ln x \to \infty \), which implies \( f(x) \to -\infty \).

The graph will start high on the y-axis as \( x \to 0^+ \) and slope steadily downward as \( x \) increases towards infinity. This behavior defines an ever-decreasing path—a characteristic identified earlier based on the derivative analysis.

Graphing doesn't just visually demonstrate the intervals of increase or decrease, but also reinforces understanding of concepts like asymptotic behavior when approaching limits. Carefully sketching, one observes no turns or bends in the function—a direct consequence of the lack of relative maxima or minima.

Thus, combining calculus with sketching techniques results in a comprehensive understanding of the function's behavior.