In Exercises 15-30 , find the absolute maximum and minimum values of each function on the given

In Exercises 15-30 , find the absolute maximum and minimum...

Key Concepts

Secant Function

The secant function is defined as the reciprocal of the cosine function. Understanding its properties, such as periodicity, symmetry, and where it is undefined due to the cosine being zero, is crucial when analyzing its behavior and graph over any given interval.

Absolute Extrema

Absolute extrema refer to the largest (maximum) and smallest (minimum) function values attained over a given domain. In optimization problems, these values identify the highest and lowest points on the graph of the function over a specified interval.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must have both an absolute maximum and an absolute minimum on that interval. This theorem provides the foundation for finding the absolute extrema by ensuring they exist within the domain.

Critical Points

Critical points of a function are the points in its domain where the derivative is zero or undefined. These points are important for optimization as they are potential candidates for local extrema, which need to be compared with the function values at the endpoints to determine the absolute extrema.

Endpoint Evaluation

When finding the absolute extrema on a closed interval, it is essential to evaluate the function at the endpoints of the interval as well as at any critical points within the interval. This comprehensive evaluation ensures that the highest and lowest function values are correctly identified.

Transcript

00:01 For this problem, we have been given a function, g of x equals secan of x, and an interval that we're going to inspect.

00:08 X is between negative pi over 3 and pi over 6 inclusive.

00:12 Now, the goal of this problem is to find both the absolute minimum and maximum values, and then we're going to graph it and see if what we expected to see with the calculus matches what we see when we graph out the function.

00:24 Okay, so let's take a step back.

00:26 Why are we expecting to see a minimum and maximum value on this interval? well, the extreme value theorem says, if you have a continuous function on a closed interval, then you will have both a minimum and a maximum value somewhere on that interval.

00:45 Well, i can tell at a glance that this is a closed interval.

00:48 I'm including both of the endpoints, negative pi over 3 and pi over 6.

00:53 But what about continuous? if i come over and look at my function, secant is the reciprocal of cosine.

01:02 Now, cosine does equal zero, so secant will have some discontinuities.

01:06 It will have some asymptotes.

01:08 Those will occur when cosine is zero.

01:11 That happens if x equals pi over two, three pi over two, and so on.

01:22 Add pi to that and you can just keep going around your circle.

01:25 The good news is none of those occur within our given interval.

01:30 So even though secant has some discontinuities, it is continuous on the given interval.

01:36 So we'll have minimum and maximum.

01:40 But now where do we look for those? there's two places we're going to examine.

01:44 First, we need to look at the end points of our interval.

01:49 And then second, we need to look at any critical points that our function has on this interval.

01:53 As a reminder, critical points occur either when our derivative of the function equals zero or the derivative is undefined.

02:06 So let's take a look at that first derivative.

02:09 The derivative of secant is secant times tangent.

02:19 Okay.

02:20 Now, is that ever equal to zero or undefined? well, some people have no problem looking at this and telling for sure what's happening.

02:28 For others, it's helpful to put things in terms of sine and cosine.

02:32 That makes them sometimes a little bit easier to see.

02:34 So you don't have to do this step, but i'm going to rewrite this derivative and put everything in terms of sign and cosine.

02:41 Secant is the reciprocal of cosine...

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