(a) find the critical numbers of f (if any), (b) find the...

(a) find the critical numbers of $f$ (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.
$$f(x)=x^{4}-32 x+4$$

Key Concepts

Graphing Utility for Function Analysis

Using a graphing utility complements analytical methods by providing a visual representation of the function. Such tools help verify the findings from derivative analysis, displaying where the function increases or decreases and highlighting potential extrema. Graphing enables a better intuitive understanding of the function's behavior and serves as a confirmatory check for analytical results.

First Derivative Test

The First Derivative Test is a method used to classify the nature of critical points by examining the changes in the sign of the derivative around those points. By determining whether the derivative changes from positive to negative or from negative to positive, one can identify local maxima or minima. This test provides a practical way to confirm the existence and type of relative extrema.

Critical Numbers

Critical numbers are the x-values in a function's domain where its derivative is either zero or undefined. These points are significant because they can indicate potential local maxima or minima, helping to analyze the behavior of the function. Identifying critical numbers is a fundamental step in applying calculus to optimization problems and understanding function graphs.

Increasing/Decreasing Intervals

Determining where a function is increasing or decreasing involves analyzing the sign of its derivative. If the derivative is positive on an interval, the function is increasing there; if it is negative, the function is decreasing. This concept is crucial because it allows one to understand the overall behavior and trends in the function, including the location of peaks and valleys.

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