# Finding Relative Extrema of g(x)=x sech(x)

• kari82
In summary, the conversation discusses finding the relative extrema of the function g(x)=x sech(x) and the difficulty in solving the equation -xtanh(x)+1=0. It is suggested to use a graph of the derivative for insight.
kari82
Hello,

can i get help with this?

Find any relative extrema of the function

g(x)=x sech(x)

First i find the derivative
g'(x)=sech(x)(-xtanh(x)+1)

set to 0

sechx=0 can't be solved

-xtanh(x)+1=0
xtanh(x)=1
I know i suppose to solve that to get the critical points and then do the sign test... but i don't know how to solve that equation.

Thanks!

Hi kari82

I don't have much experience with hyperbolic functions, so I don't know how or whether that equation can be solved analytically, but perhaps a graph of the derivative can help provide insight to your question:

http://www.wolframalpha.com/input/?i=d/dx(x*sech(x))

If you scroll down, it even tells you the roots. The fact that they used approximately equal to symbol may imply that there is no analytic solution and only a numerical one. But I really don't know for sure.

thanks! I didnt know about that website.. very useful... :-)

## 1. What is the definition of a relative extremum?

A relative extremum is a point on a graph where the function reaches a maximum or minimum value within a specific interval, but may not be the overall maximum or minimum of the entire function.

## 2. How do you find the relative extrema of a function?

To find the relative extrema of a function, you need to take the derivative of the function and set it equal to zero. Then, solve for the value(s) of x that make the derivative equal to zero. These values will be the x-coordinates of the relative extrema.

## 3. What is the formula for finding the relative extrema of a function?

The formula for finding the relative extrema of a function is:
1. Take the derivative of the function.
2. Set the derivative equal to zero.
3. Solve for the x-values.
4. Plug the x-values into the original function to find the y-values.

## 4. How do you classify the type of relative extremum (maximum or minimum)?

You can classify the type of relative extremum by looking at the concavity of the graph. If the graph is concave up, the relative extremum is a minimum. If the graph is concave down, the relative extremum is a maximum.

## 5. Can a function have more than one relative extremum?

Yes, a function can have multiple relative extrema. These points can be found by taking the derivative and setting it equal to zero for each interval where the concavity changes. However, a function can only have one absolute maximum and one absolute minimum.

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