Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=2x^4

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For polynomial functions like ƒ(x) = 2x^4, the domain is typically all real numbers, as there are no restrictions such as division by zero or square roots of negative numbers. Understanding the domain is crucial for analyzing the behavior of the function across its entire range.

Increasing and Decreasing Functions

A function is considered increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, a function is decreasing if the value at the second point is less than at the first. Identifying these intervals involves analyzing the first derivative of the function, which indicates where the function's slope is positive (increasing) or negative (decreasing).

First Derivative Test

The First Derivative Test is a method used to determine the local maxima and minima of a function by examining its first derivative. By finding critical points where the derivative equals zero or is undefined, and testing the sign of the derivative around these points, one can ascertain where the function is increasing or decreasing. This test is essential for graphing the function and understanding its overall behavior.