Use a graphing utility to graph the function and find the absolute extrema of the function on th

Use a graphing utility to graph the function and find the...

Key Concepts

Absolute Extrema

Absolute extrema refer to the highest (maximum) and lowest (minimum) values that a function attains on a specified domain. Finding these values involves analyzing the function over the entire interval, including testing both interior points (like critical points) and endpoints (if they are included in the interval).

Endpoints of Intervals (Open/Closed Intervals)

When determining extrema, it is critical to understand the nature of the interval endpoints. Closed endpoints are part of the interval and can be evaluated directly, whereas open endpoints are not included, meaning that the function may approach a value arbitrarily close to an endpoint without ever reaching it. This distinction significantly affects the existence and computation of absolute extrema.

Critical Points

Critical points of a function are where the derivative is zero or undefined, indicating potential local maxima or minima. These points are essential in the search for absolute extrema because they represent candidates where the function might achieve extreme values within the interior of the interval.

Rational Functions

Rational functions, which are expressed as the ratio of two polynomials, often exhibit behaviors such as discontinuities and asymptotic behavior. Analyzing these functions requires careful consideration of their domains and the behavior near points where the denominator is zero, as these factors influence the overall graph and extremal values.

Vertical Asymptotes

Vertical asymptotes occur when a function's value increases or decreases without bound as the input approaches a certain point, typically where the denominator of a rational function is zero. Recognizing vertical asymptotes is crucial because they mark excluded values from the domain and influence the behavior of the function near the boundaries of the interval.

Graphing Utility

Graphing utilities are computational tools that allow for visualizing the behavior of mathematical functions over specified intervals. They provide an illustrative way to identify key features such as absolute extrema, asymptotes, and discontinuities, thereby serving as an effective means for verifying analytical findings about the function.

Transcript

00:03 In this video, we are going to look at how to locate the absolute min and absolute max of the function ff x equals 2 over 2 minus x on the interval that includes the n .0 up in 2 but not including 2, if they exist.

00:17 Now recall, if we have a continuous function on a closed interval, then we are guaranteed that the absolute men and absolute max of that function both exist.

00:28 But here i have a function on a half open interval.

00:33 And there are values of x.

00:35 And specifically in this one, the function is undefined at 2.

00:40 But again, my interval doesn't include 2.

00:44 So i'm not guaranteed that the absolute min and absolute max exist, but they could.

00:49 So i still want to approach this problem looking for them in the manner that we did when they were guaranteed.

00:56 So first we want to find the critical values of f of x equals 2 over 2 minus x.

01:02 Now i can rewrite this function f of x as 2 times a quantity 2 minus x to the negative 1 power to take its derivative.

01:12 Certainly i could use a quotient rule but i could also bring up the denominator in our parentheses to the negative 1 power and use the chain rule.

01:23 So now my derivative f prime of x will the derivative of a constant times of function is the constant times the derivative of the function and the derivative of a letter base expression to a number power is the power rule so i'm going to bring down the negative one keep the base of 2 minus x subtract 1 from the exponent so negative 1 minus 1 is negative 2 and multiply by the derivative of the variable expression base so the derivative of 2 minus x is negative 1.

01:57 Rewriting that, i get f prime of x is equal to, we'll think of this as over 1.

02:03 This factor of 2 minus x to the negative 2 power goes in the denominator as 2 minus x now to the second power.

02:12 When you move it, that factor to the other part of the fraction, the sign changes on the exponent.

02:18 And then in the numerator, i have 2 times negative 1 times negative 1, so i get a positive 2.

02:24 Now, this derivative is only undefined when x is 2, but the function itself was undefined there.

02:32 So it's not a critical value plus the fact that's not in the interval.

02:36 Remember, critical values are values in the domain of the function for which the first derivative is zero or undefined.

02:44 And specifically, we're talking about the domain from zero up and two, but not including two.

02:50 So i don't have any values that are critical values from the derivative being undefined with those restrictions.

02:59 And also, my derivative cannot be zero.

03:02 If i set zero equal to two over two minus x quantity squared, the fraction is only zero when the numerator is zero and the denominator is not.

03:12 The numerator is two.

03:13 There's no values of x that would make two equal to zero.

03:16 So that doesn't give me any values.

03:19 So i don't have any critical values for this function in that interval, so there's no values to evaluate my function at.

03:29 Now remember our function is 2 over 2 minus x...

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