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

Natural Domain

The natural domain of a function is the set of all real numbers for which the function is defined. In this context, understanding the natural domain involves checking where the denominator of a rational function does not become zero, ensuring the function has a valid output. It is a fundamental step in analyzing the behavior of functions and determining potential restrictions on the inputs.

Critical Points

Critical points are the values of the independent variable where the derivative of a function is zero or undefined. They are important because they are potential locations for local extrema (maxima or minima) or points of inflection. Identifying critical points allows one to focus on regions in the function’s domain where there might be changes in the increasing or decreasing behavior of the function.

Derivative and the Quotient Rule

The derivative of a function is a measure of how the function's value changes with respect to changes in its input. When dealing with rational functions, the quotient rule is a systematic method for finding the derivative. This process involves differentiating the numerator and the denominator separately, and then combining these using the quotient rule formula. This step is crucial for finding critical points and analyzing the behavior of the function.

Local and Absolute Extrema

Local extrema (local maximum and minimum) are points where the function value is higher or lower, respectively, than those at nearby points, while absolute (or global) extrema are the highest or lowest values that the function attains over its entire domain. Finding these values involves examining the function at its critical points as well as its boundaries (if any) of the domain, and then comparing the function values to determine the most extreme outputs.

Transcript

00:01 We have a function, and we are trying to figure out where we have extrema, both local and absolute.

00:07 First, we look at this function.

00:09 It looks like there could be a possibility that the denominator would be zero at some point, which would indicate that we have a point that can't be plotted.

00:20 However, we should notice that that doesn't factor.

00:24 And if we run the discriminant test on it, which is to figure the value of b squared minus 4ac, we find that that discriminant is negative.

00:34 And if the discriminant is negative, then indicates that the quadratic formula is not going to yield any solutions.

00:43 Now, let's determine the critical points.

00:48 We can find that by taking the bottom, the derivative at least, the bottom times the derivative of the top, which is one, minus the top, times the derivative of the bottom, which is 2x plus 2, and put that over the bottom squared.

01:12 Okay, so let's simplify that top.

01:14 We're going to have x squared plus 2x plus 2.

01:20 If we foil x plus 1 times 2x plus 2, we would have 2x squared.

01:28 Middle terms of 2x and 2x, so that's plus 4x, and then 1 times 2 to get 2.

01:40 Let's go ahead and simplify that top.

01:47 We will distribute the negative that's on top.

01:55 This or the top here.

02:03 Negative x squared minus 2x, and then the plus 2 minus 2 cancels.

02:16 Now, where are critical numbers? well, the derivative would be 0 if the top is 0.

02:24 So there could be a critical number there if negative x squared minus 2x is equal to zero.

02:32 Pull out an x.

02:33 Well, actually pull out a negative x.