a. The function y=tan x+3 cot x has an absolute minimum value on the interval

Name: Problem 70, Chapter 4: Thomas Calculus
Uploaded: 2019-05-18T00:09:04-08:00
Duration: 6 min 12 s
Channel: Matt Just
Description: a. The function y=tan x+3 cot x has an absolute minimum value on the interval 0 < x < π / 2 . Find it. b. Graph the function and compare what you see with your answer in part (a).

step 1 Okay, so we have this function. F of X is tanx plus three cotangent x, and we want to show that

a. The function y=tan x+3 cot x has an absolute...

Key Concepts

Trigonometric Functions

Trigonometric functions, such as tangent and cotangent, are fundamental in the study of periodic phenomena. Their properties, including periodicity and the relationship between sine and cosine, are crucial for understanding the behavior of more complex trigonometric expressions and solving equations that involve them.

Optimization Using Calculus

Optimization involves finding the minimum or maximum values of functions. In calculus, this is typically done by calculating the derivative, setting it equal to zero to find critical points, and then analyzing these points to determine whether they correspond to absolute or relative minima or maxima, especially within a specified domain.

Graphical Analysis of Functions

Graphical analysis involves plotting a function to visually assess its behavior, such as identifying trends, estimating extrema, and understanding overall shape. This technique serves as a helpful complement to analytical methods, providing visual confirmation of results obtained through calculus and other techniques.

Inequalities (AM-GM Inequality)

The AM-GM inequality is a useful tool in optimization problems, particularly when dealing with the sum or product of nonnegative numbers. It helps establish bounds and identify conditions under which a function reaches its minimum or maximum, thereby providing an elegant alternative to derivative-based methods in certain contexts.

Transcript

00:01 Okay, so we have this function.

00:03 F of x is tanx plus three cotangent x, and we want to show that it has an absolute minimum between 0 and pi over 2.

00:27 So let's see.

00:32 So if we take the derivative, we want to find the absolute minimum, so let's find a critical point and see if it's actually a minimum.

00:40 So derivative of tangent is secant squared x, and then minus 3 co -secret squared x.

00:48 Okay, so let's find a critical point if we can.

00:52 So when does secant squared x equal three co -sacent squared x? well, let's see.

01:04 It looks like this is, i can multiply through by sine squared and i'll have tan squared x.

01:11 It's 3 or tan x plus or minus root 3.

01:17 Okay, so i know tan x is root 3 at pi over 3.

01:25 I think that's the only value that tan x is plus or minus root 3 between 0 and pi over 2.

01:30 So we have this one critical point.

01:34 And what can we do now? well, we can show, for instance, that the function is always concave up.

01:41 That would be sufficient to show that this is an absolute minimum.

01:44 So the second derivative equals 2 s squared x tanx.

01:54 This is going to be plus 6 co -sequent squared x cotangent x okay and why would this always be well so we actually see that is this always positive so let's look so what is this this is this is well we can factor it 2 out that doesn't matter and this is 1 so tan x over cosine squared x plus three times cotangine x over sine squared x.

02:52 Okay, so let's get a common denominator.

02:57 So 2, tanx, sine squared x plus 3 cotangine x cosign squared x over cosine squared x.

03:19 Okay, and so we see that this actually does have a zero.

03:37 Has one zero.

03:39 Nope.

03:40 Actually, no, it doesn't.

03:41 Right.

03:42 Because tangent of x is never, let's see...