Find the maximum and the minimum values, if any, without using derivatives of the following functions:

f(x) = –(x – 1)2 + 2 on R

We have f(x) = – (x – 1)2 + 2


It can be observed that (x – 1)2≥0 for every xR


Therefore, f(x) = – (x – 1)2 + 2≤2 for every xR


The maximum value of f is attained when (x – 1) = 0


(x – 1)=0, x=1


Since, Maximum value of f = f(1) = – (1 – 1)2 + 2 = 2


Hence, function f does not have minimum value.


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