You will use a CAS to help find the absolute extrema of the...

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where $f^{\prime}=0$. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well. c. Find the interior points where $f^{\prime}$ does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $f(x)=\pi x^2 e^{-3 x / 2},[0,5]$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0$. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and identify where they occur.
$f(x)=\pi x^2 e^{-3 x / 2},[0,5]$

Key Concepts

Absolute Extrema

Absolute extrema refer to the highest and lowest values that a function attains over a given interval. In the context of a closed interval, these values can occur either at interior points or at the endpoints of the interval. The process of finding absolute extrema involves evaluating the function at all potential candidates to determine the overall maximum and minimum values.

Critical Points

Critical points are points in the domain of a function where the derivative is either zero or does not exist. These points are essential because they are potential locations where the function changes from increasing to decreasing (or vice versa), which makes them candidates for local, and possibly absolute, extrema.

Endpoints and the Extreme Value Theorem

The Extreme Value Theorem guarantees that a continuous function defined on a closed interval will attain both a maximum and a minimum value on that interval. This means that the endpoints of the interval must always be considered when searching for absolute extrema, as they can potentially be the locations of the highest or lowest function values.

Use of Derivatives

Derivatives provide information about the slope of the function and are a crucial tool in determining where a function reaches its extreme values. Setting the derivative equal to zero or finding where it does not exist helps to identify critical points, which need to be examined to find the absolute extrema.

Function Plotting

Plotting a function and its derivative is a useful visual aid that helps to understand the behavior of the function over a given interval. It aids in identifying approximate locations of critical points and understanding the overall increase or decrease of the function, which is essential when determining the absolute extrema.

Transcript

00:01 For this problem, we've been given the function, f of x equals x to the three -fourths power minus sine of x plus one -half.

00:08 And we're looking at the interval where x is between 0 and 2 -pi inclusive.

00:13 Now, for this problem, we're going to be using a computer application to graph both the function and its derivative.

00:19 And we're going to see if we can find from that graph what is happening with our function, where are extreme values and critical points.

00:27 Now, i am using the desmos graphing application.

00:30 You can use that.

00:31 Another computer application, a computer graphing calculator.

00:35 It doesn't matter what you use.

00:37 Just make sure you're familiar with it.

00:38 You're going to want to make sure that you know how to put an interval in when you're graphing because you want to be able to clearly see those endpoints.

00:46 So i'll show you how we do that on the desmos application in a moment.

00:49 If you're using another application or a calculator, make sure you know how to do this.

00:54 You might have to research how to put that in.

00:56 In, it's going to make these problems a lot easier.

01:00 Let's take a step back.

01:02 We know that we're going to have a minimum and a maximum on this interval because of the extreme value theorem.

01:08 That theorem says if you have a continuous function on a closed interval, which we have, then you're going to have both a minimum and a maximum on that interval.

01:21 So this function has some spots.

01:25 X has to be non -negative.

01:28 It's not defined for any negative values of x.

01:31 Fortunately for us, our interval doesn't include any negative values.

01:35 So for the interval we care about, this is indeed a continuous function, so we are going to have a minimum and a maximum.

01:43 We're going to need to look at the end points and any critical points for this function.

01:52 Critical points are going to require us to find the derivative.

01:54 So let's do that right now.

01:58 The derivative, we pull down that exponent and subtract 1.

02:03 So that gives us 3 fourths x to the negative 1 fourth power.

02:07 The derivative of sine is cosine, so we'll have minus cosine of x, and the derivative of a constant is 0.

02:14 So we don't have to worry about that piece.