Find the extreme values (absolute and local) of the function over its natural domain, and where

Find the extreme values (absolute and local) of the function...

Key Concepts

Extreme Values

Extreme values of a function refer to the highest and lowest points on its graph within a given interval or over its entire domain. These include absolute (global) extremes, which are the overall maximum and minimum values, and local extremes, which are the highest or lowest values within a neighborhood of a point. Identifying these values is key in optimization problems and in understanding the function's behavior.

Local and Absolute Extrema

Local extrema are values that are either peaks (local maxima) or valleys (local minima) within a small interval around a point, while absolute extrema are the maximum and minimum values in the entire domain of the function. Determining which are absolute and which are merely local requires comparing function values at critical points and boundaries, if applicable.

Critical Points

Critical points are points in the domain of a function where the derivative is zero or undefined. These points are potential candidates for local extrema because the function's rate of change pauses or becomes discontinuous there. Finding the derivative and setting it equal to zero is a common method to identify these critical points.

Use of Derivatives in Extremum Analysis

Derivatives provide information about the increasing or decreasing behavior of a function. The first derivative test helps to determine whether a critical point is a local minimum or maximum by examining the sign changes of the derivative around that point. The second derivative test, which analyzes the concavity of the function, can also be used to confirm the nature of a critical point.

Natural Domain

The natural domain of a function is the set of all input values for which the function is defined. For polynomial functions, this typically includes all real numbers. Knowing the natural domain is important as it ensures that all potential extreme values are considered and that the analysis is conducted over the correct set of inputs.

Transcript

00:01 We're given a function and asked to find both absolute and local extrema for this graph.

00:09 I'm already going to state that there are no absolute extrema.

00:13 The reason for that is if we multiply this function out, we get an x to the fifth power.

00:20 And if we use the rules about what we know about graphs that have odd powers, any positive coefficient, we know on the far right they're going to continue to go up, and at the far left, they're going to continue to go down.

00:37 So it keeps going up and up toward infinity as x gets larger, and down toward negative infinity as x gets smaller.

00:46 So let's just focus on the local extrema.

00:51 The derivative uses the product rule, which is first times the derivative of the second, which we would use the chain rule for, plus the second times the derivative of the first, which is 3x squared.

01:10 I will take a common x squared times x minus 5 out front.

01:19 That would leave me with x times 2 for the first quantity.

01:25 And for the second quantity, it would leave me with 3 times x minus 5.

01:37 Okay, let's simplify this.

01:40 X squared times x minus five, and on the next quantity i get, if i distribute the three, i get five x minus 15.

01:54 Now, when is this equal to zero? it's going to give me three solutions.

02:06 If x is zero, x squared is zero...