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I know how to do a quadratic version but how do I find the absolute minimum and absolute maximum values of f on the given interval. $f(x) = x − \ln 2x$, $[\frac{1}{2}, 2]$

I keep getting the wrong answers

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    I am on the EVT today looking into it.2012-06-26

2 Answers 2

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The derivative is $1-\frac{2}{2x}$. It is hard to go wrong after that. The contenders are $x=1/2$, $x=1$, and $x=2$.

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First you find the critical points of $f(x)$. That would be done by finding $f'(x)$ and setting that equal to 0. So $f'(x) = 0$, solve for $x$.

After checking the critical points, you must also check the boundaries, which are $\frac{1}{2}$ and $2$.

EDIT: Also, you have to find places where $f(x)$ is not differentiable to find all critical points.

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    @ArturoMagidin: Just saw your comment. I kind of forgot about that. Thanks for raising the issue. I will add it to the response.2012-06-26