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The derivative of a function helps us understand its rate of change at any given point. For the function \( f(x) = x - \ln x \), finding the derivative is the first step in identifying critical points. It tells us where the function's slope (or the tangent line at any point) is zero, positive, or negative. In this function, we differentiate each term separately:

• The derivative of \( x \) is simply 1.
• The derivative of \( \ln x \) is \( \frac{1}{x} \).

Putting these together, the derivative of the function is \( f'(x) = 1 - \frac{1}{x} \).This derivative will allow us to find the critical values where the derivative is zero or undefined, which are the potential points where the function changes its increasing or decreasing nature.

To understand where \( f(x) = x - \ln x \) increases or decreases, we analyze the sign of the derivative \( f'(x) = 1 - \frac{1}{x} \). Critical points are identified by setting \( f'(x) \) equal to zero, resulting in a critical point at \( x = 1 \). This critical point divides the domain into intervals:

• For \( x < 1 \), substituting any value such as \( x = 0.5 \) into the derivative gives \( f'(0.5) = 1 - 2 = -1 \). This means the function is decreasing in the interval \((0,1)\).
• For \( x > 1 \), substituting a value like \( x = 2 \) yields \( f'(2) = 1 - 0.5 = 0.5 \). Thus, the function is increasing in the interval \((1, \infty)\).

Determining these intervals helps us understand the overall behavior of the function as we move along the x-axis.

Relative extrema refer to the relative maximum and minimum points of a function. These occur where the derivative changes sign, marking a transition between increasing and decreasing behavior.In our function \( f(x) = x - \ln x \), we identified that the critical point at \( x = 1 \) marks a change in direction — from decreasing (for \( x < 1 \)) to increasing (for \( x > 1 \)). This sign change in the derivative \( f'(x) \) shows that \( x = 1 \) is a relative minimum. Calculating \( f(1) \) gives us \( f(1) = 1 - \ln 1 = 1 - 0 = 1 \). Hence, there is a relative minimum point at \( (1, 1) \). This information suggests where the function reaches a low point before beginning to rise again.

With all the information gathered from the derivative, critical points, and intervals, sketching the graph of \( f(x) = x - \ln x \) becomes much easier. Here are the steps for visualizing the curve:

• Start by marking the critical point \( (1, 1) \), where we have identified a relative minimum.
• For \( x < 1 \), the slope of the function is negative, indicating the graph will descend as it approaches \( x = 1 \).
• For \( x > 1 \), the slope is positive, meaning the graph will ascend as it moves to the right.

These characteristics tell us the graph decreases to the point \( (1, 1) \) and then increases beyond it. This attaches crucial meaning to the behavior of \( f(x) \) across different values, providing a visual representation of its behavior — critical for understanding how the function behaves across its domain.