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When examining a function for critical points, the first step is to find its first derivative. This helps us determine where the function's slope is zero, indicating potential peaks, valleys, or points where the function might change behavior significantly.

For the function provided, \[ f(x) = x - b \ln x, \] we need to identify its rate of change by differentiating it.
The first derivative is found as follows:

• Derivative of \( x \) is 1.
• Derivative of \( -b \ln x \) is \(- \frac{b}{x} \) following the chain rule for logarithms.

Combining these results, we have:\[ f'(x) = 1 - \frac{b}{x}. \]

Next, we set \( f'(x) \) to zero to find critical points:\[ 1 - \frac{b}{x} = 0. \] Solving for \( x \), we find \( x = b \). Hence, the critical point occurs where the first derivative is zero, specifically at \( x = b \).

Understanding these steps forms a foundation for locating critical points and understanding how changes in the function might occur.

To determine the nature of the critical points found using the first derivative, we use the second derivative test. This test helps us understand whether a critical point is a local maximum, local minimum, or a saddle point.

The second derivative takes the rate of change of the first derivative, essentially showing how the slope itself changes:

• For the function's first derivative, \( f'(x) = 1 - \frac{b}{x}, \) we derive again to find \( f''(x). \)

The differentiation involves:

• The derivative of \(- \frac{b}{x} \) is \( \frac{b}{x^2}. \)

Thus, the second derivative is:\[ f''(x) = \frac{b}{x^2}. \]

To apply the second derivative test, we evaluate \( f''(x) \) at the critical point \( x = b \):\[ f''(b) = \frac{b}{b^2} = \frac{1}{b}. \]

Since \( \frac{1}{b} > 0, \) this positive value indicates that \( f(x) \) is concave up at \( x = b \), classifying it as a local minimum.

The second derivative test simplifies understanding complex function behaviors and assists in visualizing function curvature at various points.

Critical points categorized as local maxima or minima provide crucial insights into the behavior of a function.

A local maximum occurs when a point is higher than all nearby points, while a local minimum is lower than those surrounding it. These points don't necessarily represent the highest or lowest values of the entire function, but they do indicate significant changes.

After finding the critical point, which in our function \( f(x) = x - b \ln x \) was \( x = b \), we determined its nature using the second derivative test. We found that \( f''(x) = \frac{b}{x^2}, \) and when evaluated at \( x = b \), \( f''(b) = \frac{1}{b} > 0 \).

The positive second derivative meant that the function at this critical point is concave up, similar to how a smile appears.
Thus, the point \( x = b \) is a local minimum. In this context:

• The graph of the function would dip down, reaching its lowest at \( x = b \) before ascending, illustrating a local minimum.

Understanding local maxima and minima in functions allows us to predict function behavior, crucial in applications involving optimization or real-world problem-solving.