Find the extreme values of the function on the interval and where they occur. h(x)=|x+2|-|x-3|,

Find the extreme values of the function on the interval and...

Key Concepts

Absolute Value Functions

Absolute value functions measure the distance of a number from zero and are inherently defined in a piecewise manner. They are expressed as one formula for non-negative inputs and another for negative inputs, which is crucial when analyzing functions that incorporate absolute value expressions.

Piecewise Analysis

When a function involves absolute values, it is often necessary to break the analysis into different cases based on where the expressions inside the absolute values change sign. This technique allows one to rewrite the function in different intervals, making it easier to understand and analyze its behavior over the entire domain.

Extreme Values

Extreme values refer to the maximum or minimum values that a function attains on a given domain. In calculus, finding extreme values generally involves examining the function's behavior at critical points—where derivatives are zero or undefined—and at the boundaries of the domain. This process is essential to determine the overall range of the function.

Critical Points

Critical points are values in the function’s domain where its derivative is zero or does not exist. These points are significant because they are potential locations for local maxima, minima, or points of inflection. Identifying and analyzing these points is a central part of studying the behavior of functions, particularly when searching for their extreme values.

Transcript

00:02 In order for us to find the extreme values of this, we would only forsake the derivative of it.

00:07 But before we do that, let's go ahead and try to find what this would be as a piecewise function, because it will make it a little bit easier in the long run.

00:17 So looking at this, we're going to have to break this up into three different intervals.

00:22 One where, or one for each of where these functions here, these absolute values, the values on the inside would be positive or negative.

00:31 So that point for this one is going to be x is equal to negative 2.

00:35 Then over here, this is going to be x is equal to 3.

00:37 So these are going to be the points we're going to have to look at.

00:39 And we also know that here the derivatives are going to not be defined because for absolute values, we know at the little point it's undefined.

00:48 So that's just something to keep in mind for a later.

00:52 But we're going to have an interval first where x strictly less than negative 2 or greater than equal to it.

00:59 And then we'll have one words from negative 2 to 3.

01:02 3.

01:06 And then lastly, we'll have one where it's just larger than 3.

01:09 So x larger than 3.