How Do You Determine the Absolute Extrema of the Function ln(x)/x?

In summary: So x= arctan(-1+ sqrt(2)) is not a critical number. In summary, the derivative of cos(x)/(sin(x)+ 2) is (-sin(x)(sin(x)+ 2)+ cos^2(x))/(sin(x)+ 2)^2. If that is the derivative then you critical points are where the fraction is equal to 0 or does not exist.
  • #1
helpm3pl3ase
79
0
I have a funtion which is ln(x)/x

The derivative is 1-ln(x)\x^2

I need to find the absolute max and absolute min. Now I know I need to find the critical numbers first then plug into f(x) but I am having trouble finding the critical numbers..

x^2 = 0 so x = 0

1-ln(x) = 0 so x = e

But now Iam confused on what to do. Thanks
 
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  • #2
Also Iam not sure what to do or how to find the critical numbers of this:

-[(cos(x))^2 + (sin x)(sin x) + 2]/(sin x + 2)^2
 
  • #3
anyone??
 
  • #4
For ln(x)/x, you should try to sketch the function to see what it does. As far as I can tell, it starts at x=0, f(x) = -infinity, climbs sharply to a peak at x = e, f(x) = 1/e, and then slopes downwards towards x = infinity, f(x) tending to 0. So the only finite turning point is at x = e.
 
  • #5
So now would i just plug e in the function and see what number is produced to determine my max and min?
 
  • #6
For the first problem, x= 0 is NOT a critical number because the function itself, ln(x)/x is not defined at x= 0 but x= e is.

Also Iam not sure what to do or how to find the critical numbers of this:

-[(cos(x))^2 + (sin x)(sin x) + 2]/(sin x + 2)^2
Is that the original function or its derivative?
I ask since the derivative of cos(x)/(sin(x)+ 2) is (-sin(x)(sin(x)+ 2)+ cos^2(x))/(sin(x)+ 2)^2, similar to what you have (but not exactly equal).
If that is the derivative then you critical points are where the fraction is equal to 0 or does not exist. A fraction is 0 if and only if its numerator is 0, does not exist if and only if its denominator is 0 so you need to find where the numerator -sin(x)(sin(x)+ 2)+ cos^2(x)= 0 and where the denominator (sin(x)+ 2)^2= 0. Since sin(x) is never equal to -2, the denominator is never 0. Rewrite the first equation (and notice it is my version, not yours) as -sin^2(x)- 2sin(x)+ 1- sin^2(x)= 0 or 2sin^2(x)+ 2 sin(x)- 1= 0. I would write that as y^2+ 2y= 1 (with y= sin(x)) and complete the square: y^2+ 2y+ 1= 2 so (y+ 1)^2= 2. y= -1+ sqrt(2) and y= -1- sqrt(2). sin(x)= -1+ sqrt(2) gives x= arctan(sqrt(2)-1) but there is no x such that sin(x)= -1- sqrt(2) since that is less than -1.
 

Related to How Do You Determine the Absolute Extrema of the Function ln(x)/x?

What is the definition of absolute max and absolute min?

Absolute max and absolute min refer to the highest and lowest values of a function, respectively, over a given interval or domain. They represent the extreme points of a function and are often used to determine the overall behavior and characteristics of the function.

How can I find the absolute max and absolute min of a function?

To find the absolute max and absolute min of a function, you can use the first and second derivative tests or graph the function to visually identify the extreme points. The first derivative test involves finding critical points, where the derivative of the function is equal to zero or does not exist. The second derivative test involves evaluating the concavity of the function at these critical points to determine if they are maximum or minimum points.

What is the difference between local max/min and absolute max/min?

The main difference between local max/min and absolute max/min is that local extrema refer to the highest and lowest values of a function within a specific interval, while absolute extrema refer to the overall highest and lowest values of a function over its entire domain. Local max/min points may or may not coincide with absolute max/min points, depending on the behavior of the function.

Why are absolute max/min important in mathematics and science?

Absolute max/min points are important in mathematics and science because they provide valuable information about the behavior and characteristics of a function. They can be used to determine the overall trend of a function, identify critical points, and analyze the optimization of a system. In scientific research, absolute max/min points are often used to model and predict real-world phenomena.

Can a function have more than one absolute max/min?

Yes, a function can have more than one absolute max/min. This can occur when the function has multiple local max/min points that also happen to be the overall highest/lowest values of the function. In some cases, a function may have an infinite number of absolute max/min points, such as in the case of a constant function or a periodic function with no end points.

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