Determine the location and nature of turning point

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Discussion Overview

The discussion revolves around determining the turning point of the function \( (1 - \ln x)^2 \) for \( x > 2 \), specifically whether this point is a minimum or maximum. The conversation includes calculus concepts such as derivatives and the application of the chain rule.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant asks for help in identifying the turning point of the function and whether it is a minimum or maximum.
  • Another participant notes that \( (1 - \ln x)^2 \ge 0 \) and states that \( (1 - \ln x)^2 = 0 \) occurs when \( \ln x = 1 \), prompting a question about finding the solution to this equation.
  • A participant inquires about how to compute the derivative of the function.
  • Another participant explains the process of finding the derivative using the chain rule, detailing the steps involved in differentiating the composition of functions.

Areas of Agreement / Disagreement

The discussion does not appear to reach a consensus on the nature of the turning point, as participants are still exploring the necessary calculations and concepts without definitive conclusions.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the function's behavior and the specific conditions under which the turning point is evaluated. The mathematical steps for finding the derivative are not fully resolved.

markosheehan
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whats the turning point of (1-lnx)², x>2 is it a minimum or maximum. can someone help me
 
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Re: calculus

We have $$(1-\ln x)^2\ge0$$ and, moreover, $(1-\ln x)^2=0\iff \ln x=1$. Can you find the solution of $\ln x =1$?
 
how do you get the derivative of it
 
As for any composition of functions. The outer function if $f(x)=x^2$, so first you differentiate $x^2$, and the result is $2x$. Now our whole function is not $f(x)$, but $f(1-\ln x)$. Therefore you replace $x$ in $2x$ with $1-\ln x$ to get $2(1-\ln x)$. To finish, you need to multiply this by the derivative of $1-\ln x$.

I recommend reviewing the chain rule for computing the derivative of the composition of functions in your textbook.
 

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