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November 11th, 2015, 11:57 AM   #1
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Critical Points

Find the critical points for the function: $\displaystyle f(x)=(x^2 -2|x|)/(x-1)$

$\displaystyle f'(x) = ((2x*-2sgn(x)*(x-1))-(1*x^2-2|x|))/(x-1)^2$

If i use the derivative function above without simplifying it i seem to find critical points in x=2 and x=-2. I found these points by simply trying numbers just for fun, i did not simplify the function whatsoever.

My professor simplified the derivative function to

$\displaystyle (x^2-2x+2)/(x-1)^2 , x>0$

$\displaystyle (x^2-2x-2)/(x-1)^2 , x<0$

From this we find no critical points for $\displaystyle x>0$ and a critical point $\displaystyle x= 1-sqrt(3)$

Am i doing something wrong when i find the points x=2 and x=-2? I have tried these numbers multiple times and seem to find them correct.
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November 11th, 2015, 12:24 PM   #2
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As an example,
\begin{align*}
f'(x) &= \dfrac{(2x - 2\,\text{sgn}\, x)(x - 1) - (x^2 - 2|x|)}{(x - 1)^2}\\\\
f'(2) &= \dfrac{(2(2) - 2(1))(1) - (2^2 - 2(2))}{(2 - 1)^2}\\
&= \dfrac{2 - 0}{1} = 2 \neq 0
\end{align*}
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November 11th, 2015, 12:32 PM   #3
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Thank you, i have for some reason used sgn(2)=2 which it clearly isn't.
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