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I came across this question a while back and I found it to be challenging. I thought you could help:

Identify the critical point and determine the local extreme values of the function.

y=

-x^2-2x+4 when x is equal to or less than 1

-x^2+6x-4 when x is greater than 1

Thank you for your help!

y' = -2x-2 when x ≤ 1

= -2x+6 when x>1

The first derivative is 0 at x=-1 and 3, and undefined at x=1.

Therefore, the critical points are at x = -1, 1, and 3.

y" < 0 at x=-1 and 3, so (-1,5) and (3, 5) are maxima.

y" is undefined at x=1, but if you look at the graph you can see that it is a local minimum.

https://www.flickr.com/photos/dwread/16247792786/

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