(a) If b is a positive constant and x>0, find all critical...

(a) If $b$ is a positive constant and $x>0,$ find all critical points of $f(x)=x-b \ln x$
(b) Use the second-derivative test to determine whether the function has a local maximum or local minimum at each critical point.

Key Concepts

Second Derivative Test

The second derivative test is used to determine the concavity of a function at critical points. By evaluating the second derivative at a critical point, one can infer whether the point is a local minimum (if the second derivative is positive) or a local maximum (if it is negative). This test provides a systematic approach to classifying extrema.

Local Extremum

A local extremum refers to the points where a function reaches a local maximum or minimum value compared to its neighboring points. These points are critical in optimization problems and are typically identified by first finding the critical points and then applying tests such as the second derivative test to determine the nature of each extremum.

Critical Points

Critical points are points in the domain of a function where the first derivative is zero or undefined. They are important because they can indicate where the function may have a local maximum, local minimum, or inflection point. Identifying these points is a key step in analyzing the behavior of a function.

Derivatives

The derivative of a function measures its instantaneous rate of change with respect to its independent variable. Finding the first derivative and setting it equal to zero is a common method for locating critical points. It is fundamental in calculus for understanding the behavior and trends of functions.

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