Find the critical numbers of f (if any). Find the open intervals on which the function is increa

step 1 For this problem, we are asked to find the critical numbers of f of x equals 3 minus x times E t

Find the critical numbers of f (if any). Find the open...

Key Concepts

Critical Numbers

Critical numbers are values in the domain of a function where the derivative is either zero or undefined. These points are significant because they are potential candidates where local maxima or minima may occur. In the context of this problem, determining the critical numbers involves computing the derivative of the function and finding where this derivative equals zero.

Derivative and the Product Rule

The derivative is a fundamental concept in calculus that describes the rate of change of a function with respect to its variable. When a function is defined as a product of two functions, as in this case, the product rule is used to differentiate it. The product rule states that if a function is the product of two differentiable functions, then its derivative is the derivative of the first times the second plus the first times the derivative of the second.

Increasing/Decreasing Intervals

Determining the intervals on which a function is increasing or decreasing is accomplished by analyzing the sign of the derivative over its domain. If the derivative is positive over an interval, the function is increasing on that interval; if negative, the function is decreasing. This analysis of the derivative's sign is central to understanding the behavior of the function.

Relative Extrema

Relative extrema are points within an interval where the function attains a local maximum or minimum value. By using the first derivative test, which examines the change in sign of the derivative at critical numbers, one can determine whether a given critical number corresponds to a relative maximum (when the derivative changes from positive to negative) or a relative minimum (when it changes from negative to positive).

Graphical Confirmation

Graphical confirmation involves using a graphing tool to visualize the function and its derivative. This step is beneficial because it provides a visual representation of the function's behavior, verifying the computed critical numbers, and the intervals over which the function is increasing or decreasing. It serves as an additional check to ensure that the analytical results are consistent with the observed graph.

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