The discussion revolves around finding a local minimum of a function, specifically analyzing the behavior of its derivative. The function in question appears to be related to arctan and logarithmic expressions.
The conversation includes various interpretations of the function's critical points and the nature of the roots of its derivative. Some participants suggest that the original poster may have made an error in their analysis, while others express confusion about the existence of both a maximum and a minimum given the derivative's characteristics.
There is mention of complex roots and their relevance to real extrema, as well as the need for clarity in mathematical expressions. The discussion reflects uncertainty regarding the function's definition and the implications of its increasing nature.
How do you know that the critical point is maximum? What is the function you want to find the local minimum of?Wi_N said:Homework Statement
I set the derivative to 0. i get 1 real root a max point and 2 imaginary. How can i find that local min?
what techniques are available?
ehild said:How do you know that the critical point is maximum? What is the function you want to find the local minimum of?
How did you get the other roots? Show your work.Wi_N said:i plotted the graph. there is a min point and a max point. but only the max point is attainable through der=0
arctanx -ln(x)/6 - x/3
max point at x=1 but the min x=0.28 can't be found using der.
ehild said:How did you get the other roots? Show your work.
It would be useful for other people if you showed the way of solutionWi_N said:thank you for your help. i must be going crazy or blind but i solved it.
There are three real zeros of the derivative, but one is at negative x where the function is not defined. The OP made some mistake, and he found it an got the minimum, see Post #5FactChecker said:If there is only one real zero of the derivative (as stated in post #1), there can not be a local max and a local min (as stated in post #3). Complex derivative zeros will not add a local extreme point on the real line.
Right. I just wanted to point out that there was a mistake that was clear just from basic principles.ehild said:There are three real zeros of the derivative, but one is at negative x where the function is not defined. The OP made some mistake, and he found it an got the minimum, see Post #5