# Calculus problem: Questions about the function f (x) = - x / (2x^2 + 1)

## Homework Statement:

cannot figure out how to do this after much time spent

## Relevant Equations:

f (x) = - x / (2x^2 + 1), 0

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Mark44
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Homework Statement:: cannot figure out how to do this after much time spent
Relevant Equations:: f (x) = - x / (2x^2 + 1), 0

shown in attachmentView attachment 264392
I can see that you've done some work, but the screen shot doesn't show what you did.

What did you do to find the minimum point? Did you find the derivative, f'(x) and set it to 0?
That may or may not be the absolute minimum point. Maxima or minima can occur at places where the derivative is zero, or at endpoints of the interval, or at places where the derivative is undefined, but the function itself is defined. You have found that the function yields ##0## as the maximum when ##x=0##, the least value allowed for ##x##. For that to be so, all other allowed values of ##x## must result in the function producing quantities that are ##<0##, i.e. negative quantities.

What do you find is attained when ##x=0.1##, a low value for ##x##, or when ##x=1##, the median value for ##x##, or when ##x=1.9##, a high value for ##x##, and when ##x=2##, the greatest value for ##x##?

You might find the following article, which recounts some of the history of finding minima and maxima, to be intriguing: https://en.wikipedia.org/wiki/Adequality#Fermat's_method