Find the critical points and the intervals on which the...

Find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point yields a local min or max (or neither).
$y=e^{-x} \cos x, \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

Key Concepts

Differentiation Techniques (Product Rule and Chain Rule)

Differentiation techniques such as the product rule and chain rule are essential for finding the derivative of composite functions or products of functions. The product rule is used when differentiating a product of two functions, while the chain rule is applied when differentiating a composite function, enabling the calculation of derivatives necessary for identifying critical points and analyzing function behavior.

Increasing/Decreasing Intervals

Intervals where a function is increasing or decreasing are determined by the sign of its derivative. If the derivative is positive over an interval, the function is increasing on that interval; if negative, the function is decreasing. Establishing these intervals helps in understanding the overall behavior and trends of the function.

Critical Points

Critical points are values in the domain of a function where its derivative is zero or undefined. These points are of interest because they can indicate potential local maxima, local minima, or saddle points in the function's graph.

First Derivative Test

The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign changes of the derivative before and after the critical point, one can infer the nature of the extremum.

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