Determine where each function is increasing and where each is decreasing. f(x)=x e^x

Name: Problem 39, Chapter 4: Calculus, Early Transcendentals
Uploaded: 2021-08-03T01:24:26-08:00
Duration: 1 min 23 s
Channel: Jonathon Brumley
Description: Determine where each function is increasing and where each is decreasing. f(x)=x e^x

step 1 we wish to find where this function increases and decreases. So we're going to look at the deriv

Determine where each function is increasing and where each...

Key Concepts

Increasing and Decreasing Functions

These refer to intervals in the domain of a function where the function's values either consistently rise or fall. A function is increasing on an interval if, whenever x1 < x2, then f(x1) < f(x2). Conversely, the function is decreasing if f(x1) > f(x2) for every x1 < x2. This concept is essential in understanding the behavior of functions and is often determined using derivative tests.

First Derivative Test

This is a method used to determine the increasing or decreasing behavior of a function by analyzing the sign of its derivative. If the derivative is positive over an interval, then the function is increasing there; if it is negative, the function is decreasing. The first derivative test is also vital in locating local extrema by identifying where the derivative changes sign.

Product Rule

The product rule is a fundamental differentiation technique used when dealing with functions that are the product of two differentiable functions. It states that the derivative of a product uv is u'v + uv'. This rule is critical for differentiating functions that are formed by multiplying simpler functions, and it is particularly useful when one of the factors has well-known properties, such as the exponential function.

Properties of the Exponential Function

Exponential functions, such as e^x, have the distinctive property of being always positive for all real numbers. This property simplifies the process of analyzing the sign of derivatives when an exponential function is one of the factors, as it implies that the sign of the derivative is determined by the other factors in the expression.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are important because they can indicate potential local maxima, minima, or changes in the monotonicity of the function. Identifying critical points is a key step in analyzing the overall behavior of a function using the derivative.

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