In Exercises 17-42,( a ) find the critical numbers of f( if any ),(b) find the open interval(s)

step 1 So we have a function of x that is equal to 2 x plus 1 over x. Critical value can get here. By f

In Exercises 17-42,( a ) find the critical numbers of f( if...

In Exercises $17-42,(\text { a })$ find the critical numbers of $f(\text { if any }),(b)$ find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results. $$ f(x)=2 x+\frac{1}{x} $$

Key Concepts

Graphical Analysis

Graphical analysis involves using graphing tools or utilities to visualize the function along with its derivative. This method provides a confirmation of analytic results, allowing one to observe the function's behavior, critical points, and intervals of monotonicity directly from the graph.

First Derivative Test

The First Derivative Test is a method used to classify critical points of a function by examining the sign changes of the derivative. By determining whether the derivative changes from positive to negative (indicating a local maximum) or from negative to positive (indicating a local minimum), one can accurately identify and classify relative extrema.

Critical Numbers

Critical numbers are values within the domain of a function where its derivative is either zero or undefined. They are essential in determining where the function may experience local extrema (maxima or minima) or inflection points, serving as potential candidates for these features in the function's graph.

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input. It is a fundamental concept in calculus that helps in analyzing function behavior, such as increasing or decreasing trends, and is used to locate critical numbers.

Monotonicity (Increasing/Decreasing Intervals)

Identifying intervals where a function is increasing or decreasing involves analyzing the sign of its derivative. If the derivative is positive over an interval, the function is increasing on that interval; if negative, it is decreasing. This analysis is crucial in understanding the overall behavior and trends of the function.

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