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Advanced Math/Extreme Values


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.

-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.

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