Find absolute extrema of the function $ \ f(x)=1-|x-1| , \ \ x \in [-9,4] \ $ on the closed interval

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Find absolute extrema of the function $ \ f(x)=1-|x-1| , \ \ x \in [-9,4] \ $ on the closed interval.

Answer:

$f(x)= \begin{array} (2-x) , \ \ if \ x \in [1,4] \\ \ \ x , \ \ if \ \ x \in [-9,1] \end{array} \ $

The function has not critical point.

Thus the extrema occurs at the end points $ \ x=1, x=4, x=-9 \ $

Now,

$ f(1)=1, \ f(-9)=-9 , \ f(4)=-2 \ $

Thus,

absolute minima $ \ f(-9)=-9 \ $

absolute maxima $ \ f(1)=1 \ $

I need confirmation of my work.

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1 Answer

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Your work is correct. The absolute maximum value of $1$ is attained at $x=1$ and the absolute minimum value of $-9$ is attained at $x=-9.$

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